Time Frequency Analysis Techniques in Terahertz Pulsed Imaging

Time Frequency Analysis Techniques inTerahertz Pulsed Imaging

by

James William Handley

Submitted in accordance with the requirementsfor the degree of Doctor of Philosophy.

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ITY O

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The University of LeedsSchool of Computing

December 2003

The candidate confirms that the work submitted is his own and the appropriatecredit has been given where reference has been made to the work of others.

This copy has been supplied on the understanding that it is copyright material andthat no quotation from the thesis may be published without proper

acknowledgement.

Abstract

Terahertz (THz) radiation is abundant in the natural world yet very hard to harness in

the laboratory. Forming the boundary between ‘radio’ and ‘light’, the so called “terahertz

gap” results from the failure of optical techniques to operate below a few hundred tera-

hertz, and likewise the failure of electronic methods to operate above a few hundred giga-

hertz. However, recent advances in opto-electronic and semiconductor technology have

enabled bright THz radiation to be coherently generated and detected, and THz imaging

systems are now commercially available, if still very expensive. Terahertz pulsed imaging

data are unusual in that an entire time series is ‘behind’ every pixel of the image. While

resulting in rich data sets, this high dimensionality necessitates some form of distillation

or extraction of pertinent features before images can be formed.

Within this thesis the technology of THz pulsed imaging is examined, together with

the imaging modalities that are employed and the type of data that are acquired. The

sources of noise are categorised, and it is demonstrated that this noise can be modelled

by the family of stable distributions, but that it is neither normally distributed nor dis-

tributed according to a simple mixture of Gaussians. Joint time-frequency techniques

such as those used in RADAR or ultrasound – windowed Fourier transforms and wavelet

transforms – are applied to THz data, and are shown to be appropriate tools to use when

analysing and processing THz pulses, particularly in signal compression. Finally, cluster-

ing algorithms in time, frequency, and time-frequency based feature spaces demonstrate

that such tools have potential application in the segmentation of THz images into their

constituent regions.

The analyses herein improve our understanding of the nature of THz data, and the

techniques developed are steps along the road to move THz imaging into real world ap-

plications, such as dental and medical imaging and diagnosis.

i

Acknowledgements

I would like to thank my supervisors, Dr E Berry and Prof R Boyle, and additionally

Dr A Fitzgerald, for all the assistance and guidance they offered during my research and

writing.

I would like to thank the members of Teravision Workpackage 4; specifically CoMIR

and IMP at the University of Leeds, Physikalisches Institut, J.W. Goethe–Universitat,

Frankfurt, and TeraView (formerly Toshiba Research Europe Laboratory) in Cambridge

for providing THz data.

I would like to thank all my fellow researchers at Leeds (both past and present), es-

pecially Dr N Cohen for her input on stable distributions. Particular thanks also to those

within the Computer Vision and CoMIR research groups, for helping to create such a

productive and pleasant working environment.

My thanks also to the EPSRC for funding this research.

Finally I would like to thank my family and friends for all their prayers and support,

particularly my wife Anna, who went far beyond the call of duty in proof reading.

quia quod stultum est Dei sapientius est hominibus

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Declarations

Some parts of the work presented in this thesis have been published in the following

articles:

J.W. Handley, A.J. Fitzgerald, T. Loffler, K. Siebert, E. Berry, and R.D. Boyle, “Potential

Medical Applications of THz Imaging”, Proceedings Medical Image Understanding and

Analysis 2001, pp 17–20.

J.W. Handley, A.J. Fitzgerald, E. Berry, and R.D. Boyle, “Approaches to Segmentation

in Medical Terahertz Pulsed Imaging”, Proceedings Medical Image Understanding and

Analysis 2002, pp 157–160.

J.W. Handley, A.J. Fitzgerald, E. Berry, and R.D. Boyle, “Wavelet Compression in Medical

Terahertz Pulsed Imaging”, Physics in Medicine and Biology, 47 (21) pp 3885–3892,

2002.

J.W. Handley, A.J. Fitzgerald, E. Berry, and R.D. Boyle, “The Short Time Fourier Transform

applied to Terahertz Pulsed Imaging”, Medical Physics, 30 (6) p 1541, 2003 (abstract).

E. Berry, R.D. Boyle, A.J. Fitzgerald, and J.W. Handley, “Time and Frequency Analysis in

Terahertz Pulsed Imaging”, in B. Bhanu and I. Pavlidis, editors, Computer Vision Beyond

the Visible Spectrum, Chapter 9, pages 290–329. Springer–Verlag, in press.

J.W. Handley, A.J. Fitzgerald, E. Berry, and R.D. Boyle, “Distinguishing between Materi-

als in Terahertz Pulsed Imaging using Wide-Band Cross Ambiguity Functions”, Digital

Signal Processing, 14 (2) pp 99-111, 2004.

E. Berry, J.W. Handley, A.J. Fitzgerald, W.J. Merchant, R.D. Boyle, N.N. Zinov’ev, R.E.

Miles, J.M. Chamberlain, and M.A Smith, “Multispectral Classification Techniques

for Terahertz Pulsed Imaging: an Example in Histopathology”, Medical Engineering and

Physics, in press 2004.

iii

Contents

1 Introduction 1

1.1 Imaging with Terahertz Pulses . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Aims and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Terahertz imaging: an historical perspective 7

2.1 The Development of Terahertz Systems . . . . . . . . . . . . . . . . . . 8

2.1.1 Other Terahertz Technologies . . . . . . . . . . . . . . . . . . . 9

2.2 Terahertz Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Computer Vision . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Noise in Terahertz Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Theory 16

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Terahertz pulsed imaging systems . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 A review of time and frequency analyses . . . . . . . . . . . . . . . . . . 21

3.3.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Short Time Fourier Transform . . . . . . . . . . . . . . . . . . . 22

3.3.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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3.3.4 Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.5 Wide Band Cross Ambiguity Functions . . . . . . . . . . . . . . 30

3.4 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1 K-means Clustering . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Noise in pulsed terahertz systems 34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Noise sources in terahertz imaging . . . . . . . . . . . . . . . . . 35

4.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Estimating the noise . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 Building and fitting the model . . . . . . . . . . . . . . . . . . . 42

4.2.3 Evaluating the Model . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.4 Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Blocked scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.2 Free air scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.3 Mean pulse of step wedges . . . . . . . . . . . . . . . . . . . . . 52

4.3.4 Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Terahertz Imaging: Using Optical Parameters as a Contrast Mechanism 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 Complex Refractive Index . . . . . . . . . . . . . . . . . . . . . 69

5.1.2 Broadband Optical Properties . . . . . . . . . . . . . . . . . . . 70

5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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5.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.2 Traditional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.4 WBCAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.1 Traditional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.2 STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.3 WBCAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.4 Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.5 WBCAF Absorption Parameter . . . . . . . . . . . . . . . . . . 97

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4.1 STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.2 Simulation of short acquisition . . . . . . . . . . . . . . . . . . . 100

5.4.3 WBCAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Approaches to Segmentation of Terahertz Pulsed Imaging Data 104

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.1 Tooth Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.2 Phantoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

vi

7 Conclusions and future work 120

7.1 Summary of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Bibliography 125

A Probability Plot Correlation Coefficient Critical Values 133

A.1 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B Full results of STFT Analysis 135

B.1 Refractive Index Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.2 Absorption Coefficient Profiles . . . . . . . . . . . . . . . . . . . . . . . 143

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List of Figures

1.1 The electromagnetic spectrum, with the “terahertz gap” highlighted. . . . 2

1.2 An example terahertz pulse, acquired at Leeds. . . . . . . . . . . . . . . 3

3.1 Schematic of a terahertz pulsed imaging system in transmission mode. . . 17

3.2 Photograph of the terahertz imaging system at Leeds. . . . . . . . . . . . 19

3.3 Examples of terahertz pulses in (a) time domain, and (b) magnitude of

Fourier coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Signals presented in different domains; the time domain (top), the power

spectrum from the FT (middle), and the spectrogram from the STFT (bot-

tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Signals presented in different domains: the time domain (top), the power


tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 A signal presented in different domains: the time domain (top), the power


tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.7 The STFT of a 2.5 Hz wave followed by a 25 Hz wave, with Gaussian

window function of σ 100 s (top) and 100 ms (bottom). Note that the

viewing angle changes for clarity. . . . . . . . . . . . . . . . . . . . . . 27

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3.8 Examples of window functions — (a) Gaussian, (b) square, (c) triangular.

All are shown with equal ‘width’, and have been normalised to the same

height for ease of comparison. . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 The introduction of frequency artefacts by using discontinuous window

functions in the STFT analysis of a chirp signal. The coefficients are

shown from analyses using (a) a Gaussian window function, (b) a trian-

gular window function, and (c) a rectangular window function. . . . . . . 28

3.10 Standard k-means clustering. . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Idealised cross section of a step wedge phantom. . . . . . . . . . . . . . 41

4.2 Some example probability density functions of stable distributions. . . . . 45

4.3 Standard EM algorithm to fit a mixture of m Gaussians to n samples xi. . 46

4.4 A blocked terahertz scan (noise only) with a time constant of 500ms

shown as (a) a histogram, and (b) a normal probability plot. . . . . . . . . 48

4.5 A blocked terahertz scan (noise only) with a time constant of 50ms shown

as (a) a histogram, and (b) a normal probability plot. . . . . . . . . . . . 48

4.6 (a) Superimposed time domain of 16 pulses through free air acquired with

identical parameters, and (b) the root mean square difference between the

pulse and the mean pulse at each time point. . . . . . . . . . . . . . . . . 50

4.7 The RMS difference (noise) versus (a) electric field, (b) absolute value of

the electric field, and (c) the historical sum of the absolute values of the

electric field including the previous 2 values, making a total of 3 values

included in the sum (HAS3). Plot (d) is rescaled plot of (c), showing the

bottom left hand corner in more detail. . . . . . . . . . . . . . . . . . . . 50

4.8 Histogram of noise from a free air scan . . . . . . . . . . . . . . . . . . . 51

4.9 (a) Normal probability plot, and (b) GMM probability plot from a free air

scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.10 Stable distribution probability plot from a free air scan. . . . . . . . . . . 52

ix

4.11 Normal probability plot of noise from a step of nylon of (a) 0.3 mm, and

(b) 7 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.12 GMM probability plot of noise from a step of nylon of 0.3 mm . . . . . . 56

4.13 Stable distribution parameters against step thickness for nylon and resin

step wedges: (a) α , (b) β , (c) γ , (d) δ . Values for the blocked and free air

scans are shown for comparison. . . . . . . . . . . . . . . . . . . . . . . 57

4.14 The time series samples of terahertz pulses with spectral content inset, at

shrinkage to (a) 100 % (raw), (b) 50%, (c) 20%, (d) 10%, and (e) 5% of

DWT coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.15 Absorption coefficient profile vs. frequency for a nylon step wedge at

shrinkage to (a) 100% (raw), (b) 50%, (c) 20%, (d) 10%, and (e) 5%,

offset vertically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 A cross section of a bird’s skull imaged in Frankfurt. . . . . . . . . . . . 66

5.2 A nylon step wedge imaged in Frankfurt. . . . . . . . . . . . . . . . . . 67

5.3 A resin step wedge imaged in Frankfurt. . . . . . . . . . . . . . . . . . . 67

5.4 The head of a marzipan pig imaged in Frankfurt. . . . . . . . . . . . . . 68

5.5 (a) The raw angle of Fourier coefficient. (b) The result of unwrapping the

angle of the Fourier coefficients for the reference pulse from (a) (red), and

the three thinnest steps of nylon. . . . . . . . . . . . . . . . . . . . . . . 74

5.6 Time difference against step depth for a resin step wedge. . . . . . . . . . 78

5.7 Refractive Index vs. frequency profiles for (a) nylon, (b) resin, (c) skin,

(d) fat, (e) muscle, (f) vein, (g) artery, and (h) nerve. . . . . . . . . . . . . 79

5.8 Broadband relative transmission against step depth for a resin step wedge. 80

5.9 The natural logarithms of relative transmissions at 0.5 THz and 1.5 THz

against step thickness for a resin step wedge, with the broadband time

domain based values included for comparison. . . . . . . . . . . . . . . . 81

x

5.10 Absorption coefficient vs. frequency profiles for (a) nylon, (b) resin, (c)

skin, (d) fat, (e) muscle, (f) vein, (g) artery, and (h) nerve. . . . . . . . . . 82

5.11 Time domain of terahertz pulse and corresponding STFT coefficients for

(a) the reference pulse, and (b) a pulse through 2 mm of nylon . . . . . . 83

5.12 Time delay vs step thickness for a nylon step wedge at 1 THz. Calculated

using the STFT with a Gaussian window. . . . . . . . . . . . . . . . . . 84

5.13 Refractive index profiles for a nylon step wedge using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.14 Refractive index profiles for a nylon step wedge using the triangular win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.15 Refractive index profiles for a nylon step wedge using the rectangular

windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.16 Refractive index profiles calculated using the Gaussian windowed STFT

with width 101 � 5 (6.3 ps) of (a) a resin step wedge, (b) excised skin, (c)

excised fat, (d) excised muscle, (e) excised vein, and (f) excised artery . . 88

5.17 Logarithm of transmittance vs step thickness for a nylon step wedge at

1 THz, calculated using the STFT with a Gaussian window. . . . . . . . . 89

5.18 Absorption coefficient profiles for a nylon step wedge using the Gaussian

windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.19 Absorption coefficient profiles for a nylon step wedge using the triangular

windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.20 Absorption coefficient profiles for a nylon step wedge using the rectangu-

lar windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.21 Absorption coefficient profiles calculated using the Gaussian windowed

STFT with width 101 � 5 (6.3 ps) of (a) a resin step wedge, (b) excised skin,

(c) excised fat, (d) excised muscle, (e) excised vein, and (f) excised artery. 93

xi

5.22 The absorption coefficient profiles of fixed rectangular windowed data

from (a)-(c) a nylon step wedge, and (d)-(f) a resin step wedge. . . . . . . 95

5.23 The WBCAF coefficients of a pulse though 2 mm of nylon. . . . . . . . . 96

5.24 WBCAF time delay estimate against step-depth with best fit line for (a)

nylon, and (b) resin, both at scale 1. . . . . . . . . . . . . . . . . . . . . 97

5.25 Logarithm of the normalised WBCAF maxima at scale 1 for (a) nylon and

(b) resin step wedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.26 Relative transmission, as estimated by the WBCAF, against step thickness

for (a) nylon, and (b) resin. . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.27 WBCAF absorption parameter against scale for nylon and resin. . . . . . 99

6.1 (a) An example cross-section of a tooth, showing the enamel and dentine

areas, and (b) the allocation of regions in the synthetic data. . . . . . . . . 106

6.2 (a) A hand segmented radiograph of a real slice of tooth, and (b) the rela-

tive transmission image of the real slice of tooth at 1.38 THz. . . . . . . . 107

6.3 Photographs of the phantoms (a) ‘foot’, (b) ‘rabbit’, and (c) the partially

completed ‘THZ’ phantom. . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 An example terahertz pulse acquired in reflection mode. . . . . . . . . . . 111

6.5 The results of manual segmentation of the phantoms (a) ‘foot’, and (b)

‘rabbit’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.6 The clustering of the synthetic tooth images using k-means clustering on

(a) Time series or DWT coefficients, (b) FFT coefficients, (c) 3D feature

vector. (d) shows a failed clustering due to poor initialisation, in this

instance on the time series data. . . . . . . . . . . . . . . . . . . . . . . 114

6.7 (a) The relative amplitude of the pulse through the real tooth, and the k-

means clustering images on (b) Time series, (c) FFT coefficients, (d) 3D

feature vector. The white lines show the boundaries extracted from the

hand-segmented radiograph. . . . . . . . . . . . . . . . . . . . . . . . . 114

xii

6.8 The results of segmenting the ‘foot’ phantom using (a) the time domain,

(b) the frequency domain, (c) the DWT domain, and (d) the feature vector. 116

6.9 The results of segmenting the ‘rabbit’ phantom using (a) the time domain,

(b) the frequency domain, (c) the DWT domain, and (d) the feature vector. 117

6.10 The results of segmenting the THZ phantom image using the frequency

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B.1 Refractive index profiles for a resin step wedge using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.2 Refractive index profiles for excised skin using the Gaussian windowed

STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.3 Refractive index profiles for excised fat using the Gaussian windowed STFT.138

B.4 Refractive index profiles for excised muscle using the Gaussian windowed

STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

B.5 Refractive index profiles for excised vein using the Gaussian windowed

STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.6 Refractive index profiles for excised artery using the Gaussian windowed

STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B.7 Refractive index profiles for excised nerve using the Gaussian windowed

STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

B.8 Absorption coefficient profiles for a resin step wedge using the Gaussian

windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

B.9 Absorption coefficient profiles for excised skin using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.10 Absorption coefficient profiles for excised fat using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B.11 Absorption coefficient profiles for excised muscle using the Gaussian

windowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

xiii

B.12 Absorption coefficient profiles for excised vein using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.13 Absorption coefficient profiles for excised artery using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

B.14 Absorption coefficient profiles for excised nerve using the Gaussian win-

dowed STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xiv

List of Tables

4.1 Depth profiles of the step wedge phantoms . . . . . . . . . . . . . . . . . 41

4.2 Acquisition parameters of the step wedge phantoms . . . . . . . . . . . . 42

4.3 Distribution of noise at different time constants . . . . . . . . . . . . . . 49

4.4 Fit of stable distribution to noise at different time constants . . . . . . . . 49

4.5 Gaussian mixture model created using EM for a free air scan. . . . . . . . 51

4.6 Results for a stable distribution fitted to noise from a free air scan. . . . . 52

4.7 Depth profiles and pulse counts of the step wedges. . . . . . . . . . . . . 53

4.8 Normal probability plot correlation coefficient for nylon and resin step

wedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.9 Gaussian mixture models fitted to the noise distribution from nylon and

resin step wedges. Numbers are shown to three significant figures, except

the correlation coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.10 Stable distribution models fitted to the noise from nylon and resin step

wedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.11 Time delay and refractive index for different amounts of wavelet shrinkage. 59

4.12 Absorption coefficient (cm � 1) of nylon at various frequencies and shrink-

age levels, with measures of difference between each shrinkage level and

the raw data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1 The broadband refractive index value, calculated in the time domain. . . . 78

xv

5.2 Refractive indices calculated by traditional methods and using a WBCAF. 97

6.1 Description of the paints used to create the phantoms. . . . . . . . . . . . 108

6.2 Description of the sticker and paint phantoms. . . . . . . . . . . . . . . . 108

6.3 Teraview acquisition parameters. . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Number of mis-classified pixels when segmenting a synthetic and a real

tooth using k-means clustering on a variety of vectors. . . . . . . . . . . . 113

6.5 Percentage of misclassified pixels for the phantoms imaged in reflection

mode, using k-means clustering on three different vectors. In total there

were 9,933 ‘foot’ pixels and 9744 ‘rabbit’ pixels classified. . . . . . . . . 115

A.1 Critical Values for probability plot correlation coefficients. . . . . . . . . 134

xvi

Chapter 1

Introduction

Recent developments in semi-conductor technology have made the region of the electro-

magnetic spectrum previously known as the “terahertz gap”, shown in Figure 1.1, acces-

sible for imaging. Radiation around this frequency — 1 terahertz (THz) is 1012 cycles per

second or 3 mm in wavelength — although abundant in nature, falls between the limits

of optical and electronic technology, and had been impossible to coherently generate and

detect until as recently as the late 1980s.

This band forms a very interesting region of the electromagnetic spectrum for several

reasons, including the sensitivity of terahertz radiation to polar substances, such as water,

and its insensitivity to non-polar substances, rendering dust, plastic, and even clothes

almost transparent. Since the breakthrough in the 1980s, terahertz imaging technology

has spread rapidly, and a terahertz scanner was built at the University of Leeds as part of

the European Union “Teravision” project that ran from 2000-2003, alongside five other

academic and commercial entities across Europe1.

1For full details see the Teravision website — http://www.teravision.org/ — Last visited 27th November2003.

1

Chapter 1 Introduction

Figure 1.1: The electromagnetic spectrum, with the “terahertz gap” highlighted.

This thesis presents a number of novel analyses and analysis techniques which are

suitable for terahertz images.

1.1 Imaging with Terahertz Pulses

The technology considered exclusively in this thesis is terahertz pulsed imaging, whereby

broadband terahertz pulses are coherently generated, steered to interact with a sample, and

then coherently detected in the time domain. An example of a terahertz pulse is shown in

Figure 1.2. When a pulse such as this interacts with a sample, it undergoes changes which

are dependent on the optical properties of the sample at terahertz frequencies. Typically

2


the transmitted pulse will experience a delay, an attenuation, and a broadening, as the

different component frequencies are phase shifted, absorbed, reflected, and scattered.

Figure 1.2: An example terahertz pulse, acquired at Leeds.

A key feature of terahertz imaging is that an entire time series, such as that shown in

Figure 1.2, is acquired ‘behind’ every pixel of the terahertz ‘image’. On the one hand this

causes visualisation problems, as a small number of features must somehow be extracted

before a 2-D image can be formed. This is quite different from X-ray or MRI (for exam-

ple), when generally a single value obtained at each pixel or voxel can be simply mapped

to a greyscale value. On the other hand, the acquisition of a coherent time series enables

frequency specific phase change and attenuation to be calculated, opening up a rich seam

of data.

1.2 Aims and Motivations

Terahertz imaging is an immature technology — the main research emphasis is still firmly

on instrumentation — and there are several issues where improvements in technology or

understanding would be beneficial to its acceptance as a mainstream imaging technique:

• Long acquisition times

3


At the time of writing, the fastest terahertz system still requires up to 30 minutes to

acquire a one hundred by one hundred pixel image with a 512 sample time series.

• The high dimensionality of data acquired

A typical terahertz ‘pixel’ consists of 512 or 1,024 time samples, and obviously

some form of parametric extraction (or other dimensionality reduction) must be

carried out before an image can be formed. Even classifying or distinguishing be-

tween pixels is non-trivial in such a high dimensional space.

• Large data sets

The high dimensionality of the terahertz data immediately raises the issues of the

space needed for storage and bandwidth needed for transmission. A single terahertz

image, one hundred by one hundred pixels, would usually be of the order of 40 Mb.

• System instability and unfriendliness

Most terahertz systems are still built on large optical benches using delicate optical

components that are prone to failure and drift, as well as being extremely sensitive

to misalignment and even variation in atmospheric conditions. Just considering

the medical domain, it is unthinkable to have a system where the patient has to be

suspended in a harness over an optical bench for several hours without any motion.

Systems that are more user friendly systems are starting to appear on the market,

however.

• Resolution

Both the spatial and temporal resolution of the terahertz scanners have room for

improvement.

• Noise

As with any real world system, there is noise present in terahertz images. The

coherent generation and detection can give rise to excellent signal to noise ratios at

4


an individual pulse level, but the modality is still noisy at an ‘image’ level. These

high signal to noise ratios also tend to be reduced if acquisition times are shortened.

Clearly some of these items require electrical engineers and photonics experts, how-

ever this thesis endeavours to address some of the issues using signal processing and

image processing techniques. In particular, novel strategies for managing the noise, the

high dimensionality of data, the long acquisition times, and the large volume of data are

explored.

Terahertz data naturally lends itself to analyses based in the time or frequency do-

mains, and as the title of this thesis ‘Time Frequency Analysis Techniques in Terahertz

Pulsed Imaging’ suggests, this work has been undertaken in the time domain, the fre-

quency (or Fourier) domain, and in joint time/frequency domains such as those formed

by using short time Fourier transforms and wavelet transforms. In this way, this thesis

endeavours to take full advantage of the richness of terahertz pulsed imaging data while

providing useful analysis techniques to the terahertz practitioner.

1.3 Overview of Thesis

Chapter 2 provides a review of the history and development of terahertz imaging, together

with previous work carried out in this field. Chapter 3 contains a detailed description and

analysis of the terahertz image acquisition process, followed by a theoretical review of

the analysis techniques used throughout this thesis. The remaining chapters describe the

original work of the thesis, organised as follows :-

• Chapter 4

A detailed analysis of the noise in transmission mode terahertz pulsed imaging

is presented, and original statistically reliable models of this noise are built. A

novel exploration of how existing denoising techniques may be extended for data

compression is additionally presented.

5


• Chapter 5

The novel application of short time Fourier transforms and the wide-band cross

ambiguity function to transmission mode terahertz data is presented. Both are used

to determine the optical parameters of various materials, and are evaluated against

each other and against the traditional analysis using the Fourier transform. The short

time Fourier transform is also used indirectly to explore the effect that acquiring a

shorter time series would have.

• Chapter 6

Building on the previous chapter, the novel application of k-means clustering to

terahertz data in order to segment an image into its constituent materials is pre-

sented. The clustering is evaluated using a variety of feature vectors drawn from

the time domain, frequency domain, a discrete wavelet domain, and the domain of

derived physical features (such as absorbance at a given frequency). The evaluation

is against hand segmented images obtained in a different modality.

Finally, conclusions and future work are discussed in chapter 7.

6

Chapter 2

Terahertz imaging: an historical

perspective

Electromagnetic radiation in the terahertz band, broadly 300 GHz to 10 THz, was first

isolated in 1897 by Heinrich Rubens [56], but remained largely unexplored in the years

following. Falling on the boundary between microwave and infrared, this so-called “tera-

hertz gap” resulted from the failure of optical techniques to operate below a few hundred

terahertz, and likewise the failure of electronic/radio methods to operate above a few hun-

dred gigahertz.

Aside from the “because it’s there” line of reasoning, the terahertz band is interesting

because:

• The radiation is non-ionizing,

• The wavelength is shorter than for microwave wavelengths, with the associated

improvement in spatial resolution, while still being long enough to experience less

7

Chapter 2 Terahertz imaging: an historical perspective

of the Rayleigh scattering experienced by infrared,

• Terahertz radiation is highly sensitive to the presence of polar substances, such as

water and thus hydration state,

• ‘Dry’ non-polar substances, such as plastics, fibres, and so on, are almost transpar-

ent to terahertz radiation,

• Light weight molecules have strong emission or absorption lines in this region for

rotational and vibrational excitations, and finally,

• The universe is naturally bathed in terahertz radiation.

Recent advances in laser and electro-optical technologies have enabled bright tera-

hertz radiation to be coherently generated and detected, making this band accessible. It

should be noted that although there are exciting developments in the field of incoherent

detection and generation, such as terahertz cameras and telescopes, this thesis is solely

concerned with the coherent generation and detection of terahertz pulses and the process-

ing of the data thus acquired.

2.1 The Development of Terahertz Systems

Before the advent of these bright sources, terahertz radiation was generated either using

sources similar to those used in infrared Fourier Transform Spectroscopy, which generate

weak and incoherent radiation, or by bulky complex equipment like free electron lasers or

optically pumped gas lasers [1, 38]. The detection methods were also incoherent, record-

ing only the intensity of incident radiation using a helium cooled bolometer for example.

Unfortunately terahertz radiation is naturally present in abundance — black body radia-

tion at 300 K is at 6.25 THz — making this technology very prone to noise. Advances

in the fields of ultrashort pulsed lasers, non-linear optics and crystal growth techniques

8


have enabled these limitations to be overcome through the introduction of terahertz time-

domain spectroscopy [2, 60, 36, 61], which appeared in the late 1980s and early 1990s.

In these systems, femtosecond pulses are used to both generate a coherent terahertz wave

and to subsequently gate that wave’s detection. In this way extremely bright radiation is

coherently generated and detected, enabling systems to be several orders of magnitude

more sensitive than those using bolometric methods. A further advantage to coherent de-

tection is that it is possible to record the amplitude of the electric field in the time domain.

This opens up the possibility of using Fourier transforms, wavelet transforms, and other

time/frequency techniques in analysis.

The next step was the building of terahertz pulsed imaging systems using terahertz

time-domain spectroscopy technology, such as that reported by van Exter et al. in the

early 1990s [61], before the first real time imaging system was reported in 1995 by Hu

and Nuss [36]. The technology has now reached the stage where terahertz pulsed imaging

systems are commercially available1, although these are still expensive (in excess of UK

200,000 pounds), mainly due to the cost of the lasers. The terahertz pulsed imaging

process used to acquire the data that is used throughout this thesis is described in detail in

chapter 3.

The hardware and instrumentation side of terahertz imaging is possibly still the main

area of active research (see section 2.2, below), and there is every reason to expect acqui-

sition times to drop, resolution and signal to noise ratios to improve, and for the systems

to become cheaper and more compact.

2.1.1 Other Terahertz Technologies

This thesis is concerned solely with terahertz pulsed imaging, however other technologies

have been and are being developed in parallel with pulsed systems. Continuous wave

systems [42, 57] use monochromatic radiation that can be precisely tuned to a specific

1Teraview in the U.K. and Picometrix in the USA both sell “off the shelf” systems.

9


frequency, leading to correspondingly simpler data. Other advances include compact free

electron laser systems [28], and terahertz microscopy using near-field techniques [52].

The passive detection of incoherent terahertz radiation is a more established field —

for example the First International Symposium on Space TeraHertz Technology was held

in 1990, and the IEEE proceedings devoted a special issue to terahertz technology in

1992. The interested reader is referred to Phillips and Keene [53] in that issue for a re-

view of this field. Closer to home, passive detection of (incoherent) terahertz radiation

is also being explored, with the ‘first ever’ picture of a human hand taken in September

2002 using a terahertz camera built by ESA’s StarTiger project2. QinetiQ have also sub-

sequently demonstrated a passive millimetre wave scanner being used to detect weapons

or contraband hidden on a person’s body.

2.2 Terahertz Imaging

Terahertz imaging is still a very immature field, with the majority of research focused on

instrumentation and hardware. For example, at a recent Royal Society discussion meet-

ing [55], 14 out of the 20 posters presented ‘pure’ instrumentation research, and a further

2 were on the boundary of instrumentation and application. The oral presentations are

harder to classify, but of the 14 essentially research based presentations 4 solely dealt in

the application of terahertz technology, compared with 7 instrumentation and hardware

presentations. The remaining 3 tended to deal with both the technology and its applica-

tion. With only a handful of terahertz imaging systems around the world, and most of

them in physics or electrical engineering research groups, perhaps this is no surprise. In-

deed the long acquisition times (a 30 by 30 pixel image currently takes around 32 hours to

acquire on the Leeds system) and instability of the systems has meant that terahertz data

has been quite scarce. Of course the balance is shifting — the technology is constantly

2http://www.startiger.org/ — Last visited 25 November, 2003.

10


improving (the Teraview “TPI Scan™” system can acquire 100 by 100 pixel images in

around 15 to 30 minutes each, for example) and at the time of writing there is a terahertz

imaging clinical trial underway in a Cambridge hospital.

The imaging techniques initially tended toward being simple demonstrations of the

capabilities of the technology, so, for example, images were found by acquiring a single

time point at each pixel, leading to simple greyscale images based on amplitude. The

acquisition of an entire time series enables potentially more useful parametric images to

be generated ([34] for example). These have often been based on mature techniques from

other fields. Other acquisition methods based on terahertz pulsed imaging are also being

developed, such as dark field imaging [45]. More sophisticated analyses are emerging too,

such as tomographic imaging [26, 62] and reflection geometry imaging [19, 65]. Some

applications of these techniques are mentioned in the summary of this chapter.

2.2.1 Computer Vision

The application of computer vision techniques to terahertz pulsed imaging data is still in

its infancy. Herrmann et al. have suggested the use of “display modes”, for example using

parameters calculated from appropriate parts of the spectrum, such as those correspond-

ing with absorption or emission lines of particular molecules [34]. Loffler et al. have

demonstrated the range of parameters available for such displays [46]. These techniques

result in relatively simple images, for example false colour images where three different

parameters are mapped to the red, green, and blue components of a pixel’s colour. Mit-

tleman et al. suggested the use of wavelet based techniques [49] — an idea taken up by

Mickan et al. [48] and Ferguson et al. for denoising [25, 24]. This aspect of analysis is

covered in detail in section 2.3.

In addition to denoising, Ferguson et al. have been using multi-spectral classification

techniques to identify biological tissue [26]. In that work, chirped probe terahertz imag-

11


ing3 [8] is used, rather than the scanning pulse method used throughout this thesis, and

the results need to be interpreted in the light of this. They show that a two parameter lin-

ear predictor, such as a finite impulse response model, can distinguish between samples

of beef or bone, chicken, and air if the two parameters are used in a simple two dimen-

sional Mahalanobis classifier. They found that their parameter based classifier correctly

identified 297 out of the 300 pulses, whereas a simple feature vector based on two ob-

vious spectral features only managed 283. It should be noted that 150 of the 300 pulses

were used for training the parameter based classifier, and it is not clear how many were

used for training the spectral feature classifier. Finally they demonstrate their classifier

distinguishing between chicken and chicken bone, this time using a 5 parameter model

(and hence 5 dimensional classifier). 10,000 pulses were obtained in a 100x100 image,

and 150 of those pulses were chosen to train the classifier, 50 from each of chicken, bone,

and air. The evaluation was entirely qualitative — a false colour image showed the clas-

sification of each pixel with a photograph of the original sample displayed alongside for

comparison. This work is clearly in its early stages, although Ferguson et al. have the

advantage of large datasets on which to train the classifiers. In Chapter 6 a different ap-

proach is taken, and unsupervised clustering algorithms are applied to images of a cross

section of human tooth and to images of specially created phantoms using a variety of

feature vectors. Clustering is appropriate because the richer time series data allow more

flexibility in choice of vector, and the relative scarcity of data makes removing pulses for

training purposes unrealistic, which is why the unsupervised route is taken. The phantoms

define a ground truth against which these techniques may be quantitatively assessed.

3This is a very fast imaging technique that compromises temporal resolution in favour of speed.

12


2.3 Noise in Terahertz Data

The work to date on noise in terahertz imaging has taken one of two approaches; analysis

and evaluation of the system components (such as the lasers and detectors) [61, 14, 54]

and signal to noise ratio (SNR) [49, 70], or applying denoising techniques to try and

improve the data [25, 24]. The former work concerns itself with noise from a systems

engineering point of view, and considers signal to noise ratio on the basis of individual

pulses. In chapter 4 this thesis builds upon this work by analysing the noise present across

an entire image, and by building models of the ways in which pulses that have been passed

through nominally the same material vary. This is noise in a much broader sense, but these

models capture the actual variation that needs to be accounted for if algorithms are to be

reliable.

Ferguson and Abbott applied Donoho’s wavelet shrinkage algorithm for denoising [17],

as suggested by Mittleman in 1998 [50]. Wavelets are a natural choice of tool for denois-

ing, argued Mittleman, because of their “striking similarities” with terahertz pulses, and

a review of wavelet transforms may be found in chapter 3 of this thesis. Ferguson and

Abbott added white Gaussian noise to terahertz pulses in order to reduce the SNR. These

‘noised’ pulses were then denoised with various mother functions, and the improvement

in SNR was quantitatively measured, by comparison with the original pulse. Addition-

ally they employed a qualitative visual comparison method. Ferguson and Abbot take the

approach that ‘quick and dirty’ imaging can be cleaned up using denoising and filtering

techniques, and do indeed demonstrate that the noise they add can be significantly im-

proved using their techniques. However, in chapter 4 it is demonstrated that the noise in

terahertz data can not be accurately modelled simply using a white Gaussian process. Fur-

thermore Ferguson and Abbott make no attempt to discover how far the denoising can be

‘pushed’ (or in other words what the minimum threshold value for the wavelet shrinkage

can be) before errors are introduced. This second point has key application in the field of

data compression, and an exploration into the impact that wavelet based compression has

13


on the calculation of optical constants can be found in chapter 5.

2.4 Summary

It is sometimes said that terahertz imaging is a solution looking for a problem. Indeed

when compared to mature technologies just in the medical domain like ultrasound, X-ray,

and even newer technologies like CT scanning and MRI, it can be hard to see the need for

terahertz imaging. It is certainly true that terahertz imaging is not a panacea, and neither

will it replace these well established technologies — but it should be remembered that it

is only 8 years since Hu and Nuss reported the first real time imaging system in 1995 [36].

It is an exciting time for the field, and in terms of application areas terahertz radia-

tion has been used to characterise semi-conductors [30], assess the moisture content of a

leaf [31], to identify gases [50, 37], to discover items hidden in powder [35], and possibly

examine space shuttle tiles for defects [69]. In the biomedical domain, terahertz imag-

ing has been applied to dental tissue [11], skin and skin cancers [65, 66], and DNA [47],

amongst other things. Catalogues of the optical properties of human tissue have also been

published [4, 27]. The interested reader is referred to recent special issues of journals on

the biological application of terahertz radiation4.

Building on previous research in the signal processing and computer vision domains,

this thesis provides a selection of tools and techniques evaluated on terahertz data. This

thesis is thus a timely contribution to the field of terahertz pulsed imaging, presenting a

tool-kit of analysis techniques that are effective in this field, specifically the short time

Fourier transform in chapter 5, and clustering techniques for segmentation in chapter 6.

It further demonstrates the limitations of other techniques in the terahertz domain, such

as wavelet compression and cross ambiguity functions (chapter 5). Finally, by way of

chapter 4, a detailed analysis of the noise present in terahertz pulsed imaging is given,

4Physics in Medicine and Biology 47 (21), 2002, and Journal of Biological Physics 29 (2–3), 2003.

14


and it is shown that this noise may be modelled by distributions from the stable family.

15

Chapter 3

Theory

3.1 Introduction

This chapter forms a detailed review of the processes and techniques used in the remain-

der of the thesis. We will start with a detailed examination of the process of acquiring

terahertz data — a necessary step in understanding what terahertz data actually consist

of, and where noise and errors appear. This is followed by a review of the mathematical

techniques used in the time and frequency analysis of terahertz data, namely Fourier and

wavelet techniques. Finally there is a brief overview of clustering techniques.

3.2 Terahertz pulsed imaging systems

3.2.1 Hardware

In terahertz pulsed imaging, pulses of terahertz radiation are generated using either non-

linear optics or a dipole antenna, steered to interact with a sample, and are subsequently

16

Chapter 3 Theory

detected in the time domain using similar techniques to the generation. These pulses

are recorded via the selective amplification of a lock-in amplifier. The terahertz imager

may be set up in either transmission or reflection modality, where the detector is either

the diametrically opposite side of the sample to the transmitter (transmission mode) or

is the same side of the sample as the transmitter, being positioned in such as way as to

capture reflections from the sample (reflection mode). Figure 3.1 shows a schematic of a

terahertz pulsed imaging system in transmission mode, which is described in detail below.

Throughout this thesis systems employing non-linear optical rectification, as described

below, were used although some data were acquired in reflection mode.

Figure 3.1: Schematic of a terahertz pulsed imaging system in transmission mode.

An ultra-fast Ti:Sapphire laser emits femtosecond pulses of wavelength around 775 nm

(380 THz, in the near infrared), at a repeat frequency of around 80 MHz. These pulses

pass through a ‘chopper’, which modulates the pulse train for the lock-in amplifier. The

choppers operate at up to 5 kHz, but are typically used at around 200 Hz. The pulse train

therefore is modulated into 2.5 ms sections of pulses followed by 2.5 ms of nothing.

Each individual laser pulse is converted into a pulse of broadband radiation with spec-

17

Chapter 3 Theory

tral content ranging across the low terahertz frequencies (typically 0.5 to 5 THz, where

1 THz corresponds to a wavelength of 300 µm and a period of 1 ps) either through opti-

cal rectification using a nonlinear crystal (for example Ga:As), or through a dipole an-

tenna [20, 70]. In this way, a terahertz pulse train is generated, with 2.5 ms of terahertz

pulses (‘signal’) followed by 2.5 ms of no signal, with each ‘signal’ part containing around

200,000 terahertz pulses.

The terahertz pulse interacts with the sample in some way, and then is focused on an

electro-optical sampling (EOS) crystal, for example�110 � Zn:Te, for detection. The inci-

dent terahertz field creates an instantaneous birefringence in the Zn:Te, which is measured

with a near-infrared beam. This probe beam is circularly polarised with a quarter wave-

plate, before the birefringence modulates how elliptical this polarsation is. The difference

between the vertical and horizontal components are measured using a Wollaston polar-

ization splitting prism, and two balanced photodiodes. This balanced detector employs

common mode noise rejection by subtracting the ‘vertical’ signal from the ‘horizontal’

signal. If no modulation has occurred (i.e., no terahertz radiation has fallen on the crys-

tal) then the result of this subtraction is zero. If some modulation of the elipticity has

occurred, then the difference will be proportional to this modulation, which is in turn pro-

portional to the incident terahertz radiation. Thus the difference on the balanced detector

will be directly proportional to the radiation incident on the detector crystal. In this way

the actual electric field is recorded. In practice, the original laser beam is split to provide

both the pump beam and the probe beam.

Each laser pulse lasts only femtoseconds, whereas the terahertz pulse has a duration

of picoseconds, and certainly a pulse may have been delayed many tens of picoseconds

by its interaction with the sample. Thus the detected signal is a snapshot of the terahertz

electric field at that (femtosecond) instant. In order to obtain the electric field of the entire

pulse, an optical delay stage is used, which lengthens or shortens the path of the probe

beam compared to the path of the terahertz pulses. This varies the time point at which the

18

Chapter 3 Theory

Figure 3.2: Photograph of the terahertz imaging system at Leeds.

snapshot of the terahertz pulse is taken, and enables an entire time series to be collected.

The delay stage must obviously be positioned in such a way that a meaningful window

into the data is achieved, and this is position is known as the initial displacement – a value

which is essentially arbitrary to a given system.

The final stage is the lock-in amplifier (LIA). This is a device which selectively am-

plifies an incoming signal at a given frequency, in order to boost it above the noise. The

chopper makes the terahertz signal a square wave modulated at the chopping frequency,

whereas the general noise is not so modulated. The LIA requires a time-constant to be

set. This time constant dictates how long the LIA will spend acquiring each time-point,

and a typical value in the set-up described is 200 ms. Over 200 ms, the LIA will acquire

40 values, since it will acquire a single value from each 5 ms ‘period’. It will average all

of these to create a single value, after which the time-delay stage would typically move to

the next part of the pulse.

The terahertz pulsed imaging machine is a physical device built on an optical bench,

19

Chapter 3 Theory

using mirrors, lenses, and other optical components. The equipment used is very deli-

cate and must be positioned exactly — any misalignment causes a corruption of the data.

Poorly aligned optics, for example, cause part of the signal to be undetected. If the setup

remains unchanged throughout an experiment then this does not really cause a problem

— it is mainly relative values and ratios that are used. However it does make direct com-

parison between scans acquired at different times unreliable. Figure 3.2 is a photograph

of the optical part of the first terahertz pulsed imaging system at Leeds which shows the

complexity of the system. This system was used May 2001 to November 2003.

Finally, Figure 3.3 shows example terahertz pulses transmitted through varying thick-

nesses of nylon, and the corresponding power spectra. The pulses are very well localised

in time, and have a broadband frequency content. Notice that in the time domain the

pulses are translated, attenuated, and dilated by vary amounts depending on the thickness

of material. The power spectrum shows the pulse frequency content is centered around

1 THz, and that the higher frequency content is attenuated more by nylon than the lower

frequencies. This is also suggested in the time domain — the pulse through 1 mm is a lot

smoother, i.e., devoid of high frequency content.

Figure 3.3: Examples of terahertz pulses in (a) time domain, and (b) magnitude of Fouriercoefficients.

20

Chapter 3 Theory

3.3 A review of time and frequency analyses

3.3.1 The Fourier Transform

The Fourier transform (FT) needs no introduction as a tool for analysis, and full discussion

of it is beyond the scope of this thesis. However it is still the most important signal

processing tool, and as such warrants a brief overview. It is also a necessary background

for the understanding of the STFT.

A real valued periodic function f � t � , with period T has Fourier representation

f � t �� ∞

∑� ∞akeikωt

where ω � 2π T is the fundamental frequency and the Fourier coefficients are given by

ak � 1T

to� T

t0f � t � e � ikωtdt

This representation provides a decomposition of the function into frequency harmon-

ics, whose contribution is given by the coefficients ak.

For a non-periodic function, the Fourier Transform of f � t � , and its inverse are given

by

f � ω �� ∞� ∞f � t � e � iωtdt (3.1)

f � t �� 12π

� ∞� ∞f � ω � eiωtdω (3.2)

The Fourier transform may be discretised and applied to signals that have been dis-

cretely sampled (such as terahertz pulses). Each coefficient is a complex number whose

amplitude represents the power of that frequency component, and whose angle represents

the phase of that frequency component modulo 2π .

21

Chapter 3 Theory

The theory of Fourier series and transforms is described in more depth elsewhere

(for example [29]), and their application to image and signal processing is also described

elsewhere (for example [59]).

3.3.2 Short Time Fourier Transform

Although the Fourier transform is the standard spectral analysis technique, it performs

poorly at analysing non-stationary signals, since the frequency content is considered over

all time. The short-time Fourier transform (STFT) was therefore developed to overcome

this limitation. In STFT analysis, the signal is windowed using some window function

φ � t � before the Fourier transform is applied. The complete transform is then acquired by

translating this window along the signal, applying the Fourier transform to this windowed

signal at each location. In this way a 2-D transform is created, with values defined at trans-

lation points β and spectral points ξ . These broadly correspond to t and ω respectively,

although the correspondence is not exact due to the uncertainty principle. The STFT of a

function f � t � with respect to the window function φ � t � evaluated at the location � β ξ � is

therefore

ST FTφ f � β ξ �� ∞� ∞f � t � φ �β � ξ � t � dt (3.3)

where

φβ � ξ � t �� φ � t � β � e jξ t (3.4)

The window function has fixed time and frequency resolution, however, which is still

a shortcoming [9]. The trade-off between the time resolution and frequency resolution is

achieved through the width of the window function - the variance in the case of the Gaus-

sian window function. Narrow window functions trade good temporal resolution against

poor frequency analysis — an infinitesimal window width is time-domain analysis. Wide

22

Chapter 3 Theory

window functions trade good spectral resolution against poor temporal resolution — an

infinite window width is frequency-domain analysis (an infinite windowed STFT is in fact

a Fourier transform).

The choice of window function will also have an impact on the analysis. We might

expect a window with sharp edges — for example a rectangular window — to have dis-

continuities in the frequency domain, and these will reflect in the analysis. On the other

hand, a smooth window function – such as a Gaussian — will have a smoothing effect on

the analysis (see Figure 3.9).

Windowing the pulse using a rectangular function at a fixed β also simulates the ac-

quisition of fewer samples at the same temporal resolution, for instance acquiring only 10

samples either side of the response’s peak. If fewer recorded samples give accurate results

for the absorption coefficient, then the acquisition time can be shortened accordingly.

3.3.3 Application

Figures 3.4 – 3.6 show the application of the Fourier transform and the STFT to a variety

of test signals. These figures show the strengths and weaknesses of these two techniques.

Figure 3.4 shows the analysis of two simple sine waves; one at 2.5 Hz, and one at

25 Hz. The FT precisely locates these frequencies and the STFT identifies the centre

frequency, although the range of frequencies included is larger.

Figure 3.5 shows the analysis of two different combinations of the sine waves from

Figure 3.4. These figures show up the limitation of the FT, since the power spectrum for

both combinations looks essentially the same. The reason for this can be seen in (3.1) —

the terms of the integral are over all time. On the other hand, the STFT correctly identifies

the switch over between the 2.5 Hz and the 25 Hz section of the signal.

Finally, Figure 3.6 shows the analysis of a ‘chirp’ signal, where the frequency is lin-

early increasing between 0 Hz and 40 Hz over the 2.5 seconds of signal. This type of

signal can typically be caused by a rotating device (such as an engine) starting up and spin-

23

Chapter 3 Theory

2.5 Hz Sine Wave 25 Hz Sine Wave

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.20 0.40 0.60 0.80 1.00

Time (s)

Sig

nal (a

rb)

Time (s)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.20 0.40 0.60 0.80 1.00

Sig

nal (a

rb)

Frequency

(Hz)

Time (s)

Frequency

(Hz)

Time (s)

Figure 3.4: Signals presented in different domains; the time domain (top), the powerspectrum from the FT (middle), and the spectrogram from the STFT (bottom).

ning up to speed. The FT correctly identifies that there is a range of frequencies present,

but nothing beyond that. The STFT on the other hand shows the frequency changing

linearly.

The effect of changing the width of the STFT windowing function can be demon-

strated using the signal with the 2.5 Hz wave followed by the 25 Hz wave. Figure 3.7

shows the signal being analysed with both a relatively wide window (the top figure) and a

relatively narrow window (the bottom figure). Note that the viewing angle of the surface

has changed between the two figures in order to highlight the differences. With the wide

24

Chapter 3 Theory

2.5 Hz + 25 Hz Sine Waves 2.5 Hz then 25 Hz Sine Wave

Time (s)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.20 0.40 0.60 0.80 1.00

Sig

nal (a

rb)

Time (s)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.50 1.00 1.50 2.00 2.50

Sig

nal (a

rb)

Frequency

(Hz)

Time (s)

Frequency

(Hz)

Time (s)

Figure 3.5: Signals presented in different domains: the time domain (top), the powerspectrum from the FT (middle), and the spectrogram from the STFT (bottom).

window, the frequency resolution is very good, but the time resolution is poor, making it

very hard to tell when the signal switches frequency. On the other hand, with the narrow

window, the time resolution is excellent, and the distinction between the frequencies is

obvious. However the frequency resolution is very poor, providing very little information

about the frequency content.

The other parameter to consider is the window function. Figure 3.9 shows the effect

of analysing the chirp from Figure 3.6 with a Gaussian window, a triangular window,

25

Chapter 3 Theory

and a rectangular window. For examples of these window functions, see Figure 3.8. The

rectangular window function very clearly shows discontinuities in the frequency domain,

the triangular window has them to a lesser extent, and the Gaussian window does not have

any such artefacts. Note in these figures the intensity denotes the magnitude of the STFT

coefficients, with black showing the largest magnitude, and white the smallest.

0 to 40 Hz linear chirp

Time (s)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.50 1.00 1.50 2.00 2.50

Sig

nal (a

rb)

Frequency

(Hz)

Time (s)

Figure 3.6: A signal presented in different domains: the time domain (top), the powerspectrum from the FT (middle), and the spectrogram from the STFT (bottom).

26

Chapter 3 Theory

Figure 3.7: The STFT of a 2.5 Hz wave followed by a 25 Hz wave, with Gaussian windowfunction of σ 100 s (top) and 100 ms (bottom). Note that the viewing angle changes forclarity.

Figure 3.8: Examples of window functions — (a) Gaussian, (b) square, (c) triangular. Allare shown with equal ‘width’, and have been normalised to the same height for ease ofcomparison.

27

Chapter 3 Theory

Figure 3.9: The introduction of frequency artefacts by using discontinuous window func-tions in the STFT analysis of a chirp signal. The coefficients are shown from analysesusing (a) a Gaussian window function, (b) a triangular window function, and (c) a rectan-gular window function.

28

Chapter 3 Theory

3.3.4 Wavelet Transforms

The continuous wavelet transform (CWT) of a 1-D function x � t � with the mother function

ψ is defined as

CWTψ x � τ σ �� 1�σ

x � t � ψ �� t � τ

σ � dt (3.5)

where τ and σ are the translation and scale parameters respectively. This corresponds

to a correlation between the input signal and scaled/translated versions of the mother

function. In this way a 2-D plot in translation/scale space is obtained, with translation

being directly related to time, and scale being inversely related to frequency. It is not

possible to define an exact relationship between translation and time, because each trans-

lation value actually corresponds to a time window, the width of which is dependent on

scale. Similarly each scale value corresponds to a frequency window, the width of which

also depends on the scale. Thus translation corresponds to a range of times, and scale

corresponds to a range of frequency. This is an inevitable consequence of uncertainty,

and the strength of wavelet analysis lies in the optimisation of these window widths. The

centre of these windows is known however, as the centre of the time window is directly

proportional to translation, and the centre of the frequency window is inversely propor-

tional to scale. The width of the time window is directly proportional to scale (i.e., high

frequencies, which correspond to small scales, are well localised in time), whereas the

width of the frequency window is inversely proportional to scale (i.e., low frequencies,

which correspond to large scales, are well localised in frequency).

Discrete Wavelet Transform

The CWT is straightforward to discretise, for implementation on a computer, however an

efficient transform called the Discrete Wavelet Transform (DWT) may also be used.

The DWT has a similar expansion to the Fourier transform, defined as

29

Chapter 3 Theory

f � t �� σ jσka j � kφ j � k � t � (3.6)

where j and k are integers and the functions φ j � k � t � are the wavelet basis functions —

these usually form an orthoganal basis. a j � k are then the DWT coefficients of f � t � , and

are calculated using

a j � k � f � t � φ j � k � t � dt (3.7)

As with the CWT, the wavelet basis functions are a two-parameter family of functions

related to a mother function thus

φ j � k � t �� 2 j � 2φ � 2 jt � k � (3.8)

k and j are called the translation and dilation parameters respectively, and so the

wavelet basis is obtained from a single mother function through translating and scaling.

3.3.5 Wide Band Cross Ambiguity Functions

In the case of terahertz pulsed imaging, however, the interest is in the relative time-delay

and spectral changes of a pulse compared with its reference pulse. We can extract these

relative differences by using a cross ambiguity function between the sample pulse and the

reference pulse [68, 64]. The wide-band cross ambiguity function (WBCAF) is defined

as

W BCAFx1x2 � τ σ �� 1� �σ� ∞� ∞

x2 � t � x �1 � t � τσ � dt (3.9)

where x1 � t � is the reference waveform, and x2 � t � is the delayed and attenuated sam-

ple waveform. The similarities between this and the CWT of (3.5) are clear — we are

effectively using our reference pulse as the mother function. It has been noted [49] that

30

Chapter 3 Theory

terahertz pulses exhibit similar properties as wavelets, namely the compact support and

broadband content. The basis of the WBCAF will be non-orthogonal, and hence the

transform will have redundancy. However we are interested in small changes to the scale

parameter, so an orthogonal basis, which typically use dyadic scales, would not provide

the resolution of scale that was hoped for.

In practical terms, the different scales are achieved by re-sampling the reference pulse,

through digital interpolation, filtering, and decimation [13]. The filtering step is necessary

to prevent aliasing and other artefacts as the sample rate is modified, and is simply a low-

pass digital filter.

3.4 Clustering

Clustering is the technique of grouping ‘similar’ n-dimensional data points together in

order to partition a dataset. Clustering techniques fall broadly into two categories —

supervised and unsupervised.

Supervised clustering uses extensive training data to create the clusters, and subse-

quent data points are allocated to the most appropriate groups. With large data sets and

good exemplars, supervised clustering is the best method to use, and examples range from

simple linear discriminators, such as applied by Ferguson et al.to terahertz data [26], to

trained artificial neural networks such as back propagation [22] and support vector ma-

chines [12].

Unsupervised clustering, on the other hand, is appropriate where either there is in-

sufficient data for training, little a priori knowledge, or where the outcome (number of

groups, exemplars, and so on) is not known in advance. In this approach ‘similar’ pixels

are grouped together to create the clusters by minimising a cost function (the distance of

each data point from its cluster’s centre, for example). Examples of unsupervised cluster-

ing include k-means [33] and self organising networks [22, 43]. Variations include region

31

Chapter 3 Theory

merging and region splitting [33]. Of these, k-means clustering is perhaps the most well

understood and established algorithm, and has been applied to terahertz pulsed imaging

data in Chapter 6.

3.4.1 K-means Clustering

The n–dimensional data are considered to be vectors that form points in n–dimensional

space. A standard distance metric, such as Euclidean distance, is used to assess similarity,

where a small distance between two data points indicates a large similarity. Each cluster

is typically described by the location of its centroid in the n–dimensional space, one such

measure being the mean of the position of all its members. In k-means clustering clusters

are formed by initialising the desired number of cluster centroids and then adding each

data point to its ‘nearest’ centroid (which in turn updates the position of the centroid),

until all the pixels have been clustered. The k-means algorithm is shown in Figure 3.10.

This algorithm is deterministic for a given initialisation.

1. Initialisation: Use some initialisation metric to establish the ini-tial positions of each cluster.

2. Allocate points to cluster: Allocate each data point to the cluster that is ‘near-est’.

3. Update the cluster positions: Calculate the new position of each cluster based onthe mean position of all that cluster’s members.

4. Repeat until finished: Repeat 2 and 3 until the stopping criteria are met.Examples of stopping criteria are ‘no more pointschange cluster’ or ‘no cluster centroid moves bymore than a threshold’.

Figure 3.10: Standard k-means clustering.

When the n–dimensional space is formed from domains which are not directly com-

parable, care must be taken to ensure that differences in scale or units do not bias the

outcome. In this case, feature vectors should be normalised to be uni-variate within a unit

hypercube [33].

32

Chapter 3 Theory

3.5 Summary

In this chapter mechanics of terahertz pulsed imaging have been reviewed in some detail,

which will be of use throughout this thesis, but especially in chapter 4 when consider-

ing the noise present in terahertz data. We have looked at time/frequency analysis tech-

niques, starting with traditional Fourier transforms, moving through short time Fourier

transforms, and then reviewing wavelet transforms and cross ambiguity functions. These

techniques are used in chapter 5 in the interpretation of terahertz pulsed imaging data.

Finally, the areas of clustering were reviewed, specifically k-means clustering which is

applied to terahertz pulsed data in Chapter 6.

33

Chapter 4

Noise in pulsed terahertz systems

4.1 Introduction

Undesired data and the corruption of signals, or ‘noise’, are issues that affect all ‘real

world’ signals, and all signal processing applications must take it into account. However,

before the noise can be reliably removed or allowed for, it must be characterised or mod-

elled. Terahertz pulsed imaging is no exception to this rule, and in this study I estimated

the noise present in a number of terahertz data-sets and found distributions that model the

underlying distribution of the noise. The parameters of the distribution not only charac-

terise the noise but could also aid the design of algorithms that either remove the noise

or are robust against it — a process that requires good noise models [3, 6, 67]. A noise

model may also enable synthetic images to be created with realistic simulated noise.

In this chapter distributions to model the noise in pulsed terahertz imaging are empir-

ically built, with the aim of providing a model that may be used in both signal processing

and in the creation of realistically noisy synthetic terahertz images. After a brief analysis

34

Chapter 4 Noise in pulsed terahertz systems

of the theoretical noise sources, the noise is extracted from a number of terahertz data

sets that demonstrate different noise characteristics from each other, before finding the

distribution that best describes the noise in each case. I am exclusively considering noise

in the terahertz time series, and not image noise.

Finally I consider the wavelet shrinkage denoising algorithm [17] applied to terahertz

imaging [25, 24], assessing what level of denoising may be applied before statistically

significant errors are introduced in the optical parameter calculations. More details on

calculating the optical parameters may be found in Chapter 5.

4.1.1 Noise sources in terahertz imaging

In order to break down the noise components of terahertz pulsed imaging and aid the

analysis, the noise can be split into two categories, signal noise and imaging noise.

Signal noise has three main components; Johnson noise, shot noise, and set–up noise,

where set–up noise is that which is due to the atmospheric conditions, sample type, back-

ground radiation, set up of the machine, and so on. The signal noise is concerned with

noise that has deformed the terahertz pulse in some undesired way, as opposed to noise

which is an artefact of the imaging process. These are all discussed in more detail below.

Imaging noise, on the other hand, is noise that is introduced through the imaging

process. For example the incoming analogue signal is sampled onto the computer, a

process that invariably results in a digital approximation of the original analogue signal.

Similarly, signal processing or rounding errors in the computers will introduce further

errors. Finally when an entire image is being formed, if the raster scanner is misaligned

this will result in pixels in the final image being in the ‘wrong place’. Of course the

signal noise will impact the generation of images — a large amount of shot noise may be

apparent in attenuation plots, for example — but I am not concerned with these effects of

noise in this study.

35


Johnson noise

Johnson noise is due to the electron equivalent of Brownian motion. To quote Demir [15];

“Random fluctuations in the short-circuit terminal of an arbitrary linear

resistor, having resistance R and held at temperature T, are independent of all

other parameters, and can be modelled with a wide sense stationary stochastic

process with a spectral density given by

Si � f �� 2kTR

(4.1)

where k is Boltzmann’s constant.”

The book further states that

“It can be shown that a thermal noise of a linear resistor as a white wide

sense stationary stochastic process with spectral density (4.1) is accurately

modelled by a Gaussian process, ... , as a direct consequence of the central

limit theorem.”

This noise is present even when the laser is switched off. There is also background

noise present, due to the abundance of incoherent terahertz radiation — black body ra-

diation at 300 K is at 6.25 THz. However, the gated and coherent nature of the terahertz

receiver makes it reject background noise, meaning this noise is a factor of 7 times smaller

than the Johnson noise [61].

Shot noise

Existing studies have categorised the lasers and their noise sources [14, 54], and the main

noise source that results from the laser generation process is shot or quantum noise [61].

When the photons of the laser hit the detector, they give rise to a constant photocurrent,

as if a bias voltage had been applied. The mechanism for this is the induction of a charge

36


transfer and hence random effective voltage in the photo-diodes of the detector. However,

since the interactions of the photons (including their emission and detection) are stochastic

processes, there will also be a ‘quantum mottle’ effect, leading to random fluctuations in

the charge transfers. This noise is about the same order of magnitude as the Johnson

noise, and can be modelled using a wide sense stationary Gaussian process [15, 61].

Other sources of noise specifically concerned with the laser include laser timing jitter

and pulse energy fluctuation [14, 54], but these have an insignificant impact on the signal

compared with the shot noise.

Previous studies have shown that the dominant noise source in these categories is the

shot noise, giving a signal to noise ratio in the order of 100:1 [49].

Set–up noise

There are many sources of physical noise in terahertz imaging. Background radiation has

been mentioned above, but in addition to this atmospheric conditions have a large impact

on the signal acquired. The main factor of this is humidity, because terahertz radiation

is so strongly absorbed by water molecules. Water molecules also absorb and re-emit

radiation at this frequency. This type of activity is not noise in the traditional sense as it

is deterministic if the conditions do not change. However varying conditions would make

direct comparisons unreliable between pulses acquired on different days or over a long

period of time.

If a sample is present, there will also tend to be a scattering effect. There are many dif-

ferent forms of scattering (Rayleigh, Raman, Brillouin, and so on) depending on the type

of molecule, and the wavelength of the radiation. The effect this has on terahertz pulses

is two-fold; firstly it introduces a loss as some radiation will be scattered away, and sec-

ondly some spectral content may end up phase shifted, as it takes a longer route through

the material. The modelling of scatter of terahertz pulses through various materials is

an ongoing research topic, but for the purpose of this analysis it is ignored. Techniques

37


such as Monte Carlo modelling are both complex and computationally expensive, and the

empirical nature of this investigation will automatically include the effects of scattering.

The other form of imaging noise arises from losses due to absorption and reflection.

Absorption and reflection are what make imaging possible (if the radiation didn’t interact

with the material, there would be nothing to see), however inadequately allowing for

reflection losses can cause errors in other calculations, such as when deriving the optical

properties of the sample.

The main impact these losses have on this study is that a decreased level of radiation

will be transmitted through the material. This will tend to decrease the signal to noise

ratio, making it harder to be certain about what constitutes signal and what constitutes

noise. Similarly the balance of the noise component will be shifted towards noise that

is dependent on the detection technology and away from noise caused by the generation

process — this latter noise will tend to be modulated along with the signal.

Imaging noise

As discussed in Chapter 3, the terahertz pulsed imaging machine is a very delicate system

requiring careful optical alignment. Inevitable variations in the optics make direct com-

parisons between machines unreliable, and indeed scans acquired on different days may

not be directly comparable. Similarly, the lock-in amplifier can drift over time, causing a

d.c. bias to be introduced in the data acquired.

At an imaging level, as opposed to the pulse level, raster scanning errors can cause

straight edges to appear staggered. The motors moving the sample may not displace it

correctly, leading to a varying physical distance between pixels on the image.

It is expected that all these noise sources are insignificant and rectified as far as possi-

ble during the set-up and calibration of the scanner.

Finally, as the analogue waveform is captured to and processed on a computer, there

is the potential for the introduction of further errors. Similarly computers have only finite

38


accuracy on mathematical operations, and each time an operation is performed some ac-

curacy may be lost. These are both unavoidable. It is assumed that appropriate sample

rate, dynamic range, and accuracy values are chosen, but these still need to be considered

when giving an error margin.

4.1.2 Summary

In summary, the main theoretical sources of pulse noise in terahertz imaging are Johnson

noise and shot noise, both of which may be modelled as wide sense stationary Gaussian

processes.

However the additional noise sources make direct comparisons between scans unreli-

able, and the effects of scattering and atmospheric conditions may contribute significantly

to the noise levels, altering the underlying noise distribution from the simple Gaussian

processes.

4.2 Methods

The noise models were built in a three phase process. Firstly noise was estimated from

measured signals. Secondly various models were fitted to these noise estimates, using

standard distributions. Finally each model was evaluated to see how good a fit was ob-

tained, in order to assess if it indeed modelled the distribution. It was hoped that a generic

model for noise would emerge through this process.

4.2.1 Estimating the noise

The noise was estimated using three different methods. The first was simply to block all

the terahertz radiation from falling on the detector crystal. In this way, the entire signal

recorded was taken as noise, as there was no terahertz radiation present. These pulses

were acquired at the University of Leeds using a displacement of 22.5 mm, acquiring 256

39


time points, with a separation between time points of 80 fs. The time constant was set at

50 ms, 200 ms, and 500 ms in order to acquire three pulses. The LIA delay was twice the

time constant in these instances. The variation in LIA time constants was to explore the

effect that this setting has on noise.

The second method was to record free air scans several times under identical con-

ditions. With no sample in place, any variation between the scans is entirely due to re-

ceiver, transmitter, and ‘set–up’ noise such as atmospheric conditions. Sixteen pulses

were recorded at Leeds through free air using identical acquisition parameters within a

short space of time. The displacement was 22.5 mm, and again 256 steps were acquired

at 80 fs separation. The time constant was 100 ms, with a LIA delay time of 300 ms. The

mean pulse in the time domain was calculated and assumed to represent a pulse free of

random noise. The differences in the time domain between each of the pulses and the

mean pulse was then used to create the noise distribution.

It was noticed that there appeared to be a correlation between the amplitude of the

electric field and the level of noise in the free air scans. This was explored by plotting

the root mean square difference between the mean pulse and the individual pulses at time

t against various derivatives of the amplitude of the mean pulse at time t. Firstly the

amplitude was used, followed by the absolute amplitude. Finally an ‘historical absolute

sum’ over n (HASn) was used, where

HASn � t �� t

∑i �� t � � n � 1 �� meanpulse � i � �� (4.2)

The HAS was chosen as a metric because the detector cannot predict the future (what the

next value of the electric field will be), but it might ‘remember’ the recent past (what the

previous value, or values, were.) The value over which to sum, n, was chosen empirically

based on which gave the best correlation.

The final method of estimating the noise was similar to the previous method, except

40


that the pulses were recorded through homogeneous regions of an object. Two step wedge

phantoms were manufactured — one ‘nylon’ and one ‘resin’ — with the step depth pro-

files shown in Table 4.1. The nylon wedges were made from Duraform polyamide (nylon

12) by the selective laser sintering process. The resin wedges were made by stereolithog-

raphy in SL 5190 photopolymer resin from 3D Systems1. The wedges were made at

REACT, Rotherham in Jan/Feb 2001. Figure 4.1 shows an idealised cross section of one

of these wedges.

Nylon Resin0.3 mm 0.1 mm1 mm 0.5 mm2 mm 1.5 mm4 mm 3 mm6 mm 5 mm7 mm 7 mm

Table 4.1: Depth profiles of the step wedge phantoms

Figure 4.1: Idealised cross section of a step wedge phantom.

These phantoms were imaged in transmission modality on the pulsed terahertz imag-13D Systems were supplied by Ciba Geigy at the time. Ciba Geigy have subsequently been taken over

by Vantico (www.vantico.com ).

41


ing systems at Physikalisches Institut, J.W. Goethe-Universitat, Frankfurt. Table 4.2

shows the acquisition parameters. The reference pulse in both cases was a terahertz pulse

through free air.

Phantom Nylon ResinSpacing of time points (fs) 150 200

Number of time points / pixel 128 128Displacement -32.5 -34

Number of pixels (x � y ) 100 � 4 100 � 8x range (mm) -1 to 24 -2 to 23y range (mm) -3 to -1 0 to 4

Table 4.2: Acquisition parameters of the step wedge phantoms

A region of interest was then defined within each step, taking care to avoid the edges

to minimise boundary effects, and the pulses within that region were assumed to have

passed through the same thickness of homogeneous material. Around 40 terahertz pulses

were then recorded through each region of interest of the nylon step wedge, with around

80 recorded similarly through the resin step wedge.

The average pulse from within each region was calculated, as before, and used as an

estimate of a pulse free of random noise.

4.2.2 Building and fitting the model

The theoretical noise analysis suggests that the noise sources are wide sense stationary

processes, and therefore should be distributed in a zero-mean approximately normal dis-

tribution. These types of distribution belong to the stable family, an infinite set that in-

cludes Gaussian, Cauchy, Levy, and Gaussian mixtures [23, 39]. Broadly speaking the

family of stable distributions are heavy tailed, skewed, or both heavy tailed and skewed.

In each case the first step was to plot the noise histogram whereby the approximate

normality of the distribution could be visually confirmed.

42


Stable Distributions

Stable distributions are probability distributions that allow skewness and heavy tails, and

were characterised by Paul Levy in the 1920s. The family of distributions is exclusively

such that if random variables x and y have stable distributions, then the random variable

x y will also have a stable distribution. The Gaussian distribution is a special case of

this family, as it is the only such distribution with a finite variance [23]. There are no

closed formulae for densities and distribution functions for all the stable functions except

Gaussian, Cauchy, and Levy, which make them hard to use in practice. However there are

now computer programs to compute stable densities, distribution functions and quantiles.

I used Nolan’s ”stable” program in this work [51].

As stable distributions generally have no closed form, they are typically expressed

through the characteristic function of a stable random variable X. In this representation,

four parameters are required to describe a general stable distribution: α , β , γ , and δ . α !� 0 2 " is the index of stability, β !$#%� 1 1 " is the skewness parameter (where β � 0 means

the distribution is symmetric), γ & 0 is the scale parameter, and δ ! ℜ is the location

parameter. γ and δ are the stable equivalent of the standard deviation and variance of a

Gaussian distribution. The shape of the probability density function is determined by α

and β , and certain values give rise to specific distributions within the family, namely

α � 2 β � 0 give Gaussian distributions,α � 1 β � 0 give Cauchy distributions, andα � 1

2 β � 1 give Levy distributions.

A stable random variable X has the characteristic function

E�eiuX �'� ()* )+

e � γα , u , α � 1 � iβ � tan πα2 � sign � u �-� � iδu � α .� 1 �

e � γα , u , α � 1 � iβ 2π � sign � u �-� ln � , u , �� iδu � α � 1 � (4.3)

43


where

sign � u �� ())))* ))))+� 1 � u / 0 �0 � u � 0 �1 � u & 0 �

Figure 4.2 shows some examples of stable distributions, together with the effects of

varying the parameters.

Fitting the model

The stable family of distributions is difficult to use analytically, so before fitting a stable

distribution the noise was first tested against a Gaussian model and a Gaussian mixture

model, or GMM — a model built from the simple addition of two or more Gaussian

distributions.

The noise distributions were all well sampled, and so the mean and standard deviation

of this noise were used for the parameters of the hypothetical Gaussian distribution.

The Expectation Maximisation (EM) algorithm [16], as shown in Figure 4.3, was

used to build the GMMs. This is a two stage iterative process that calculates the local

maximum likelihood fit of a given arbitrary number of Gaussians to a data set. Although

EM is sensitive to initialisation in many applications, in this context the distribution being

fitted is a heavy tailed normal distribution, which means the Gaussians that fit it will have

approximately the same mean. This in turn means the fit will not particularly depend on

the initialisation metric. The Gaussians were initialised to have equal weighting and to

be centered on the mean of the data. The standard deviation of the first Gaussian was

initialised to the standard deviation of the data, and subsequent Gaussians initialised with

increasingly larger standard deviations. EM can also have difficulties with sparse data sets

causing singularities – however in this application the data are not sparse so there is no

problem here. The algorithm stops when the parameters of the Gaussians remain constant

for an iteration.

44


Figure 4.2: Some example probability density functions of stable distributions.

An alternative approach to building GMMs is to use a ‘kernel method’ [58], whereby

each data point is modelled by a single Gaussian. However this approach is computa-

tionally expensive, and while it would provide a good model for the noise, it would not

provide insight into the underlying noise distributions.

45


E-step: Compute the contribution of the ith sampleto the jth Gaussian.

pi j =w jN � xi:µ j � σ2

j �∑M

k 0 1 wkN � xi:µk � σ2k �

M-step: Compute the parameters of the Gaussians.

w j = 1n ∑n

i � 1 pi j

µ j = ∑ni 0 1 pi jxi

∑ni 0 1 pi j

σ 2j = ∑n

i 0 1 pi j � xi � µ j �1� xi � µ j � T∑n

i 0 1 pi j

Figure 4.3: Standard EM algorithm to fit a mixture of m Gaussians to n samples xi.

4.2.3 Evaluating the Model

Whichever model is derived, the result is a distribution with known parameters. This

hypothetical distribution needs to be evaluated to see if it indeed models the noise from

which it was derived. To test the model, a null hypothesis, H0, was created that stated the

data are sampled from this distribution of known parameters.

A probability plot was plotted for each of the models — the Gaussian, GMM, and

stable distributions. The expected values from the model were plotted against the actual

data values, and where the model is accurate a straight line of gradient 1 is produced.

The correlation coefficient between the expected values and the actual values gives the

probability plot correlation coefficient, PPCC [7].

A significance value of 0.05 (5%) was chosen, and the PPCCs were then compared to

the critical value of the PPCC for that significance. Appendix A shows the critical values

for different sample sizes for the 0.05 and 0.01 significance values. If the PPCC is greater

than the critical value, then H0 (that the data came from the distribution specified) can

be accepted. The probability plot itself also gives a graphical indication of how the data

distribution varies from the hypothetical model.

46


4.2.4 Denoising

The nylon step wedge data from Frankfurt was used in the denoising analysis. The data

were denoised by transforming each waveform into its DWT coefficients, and setting to

zero the smallest p% of coefficients. The error this process introduced was measured

by reconstructing the waveforms from the truncated DWT coefficients, then recalculat-

ing the broadband refractive index and absorption coefficient analogues using the process

described in Chapter 5. In the case of the refractive index, the value can be directly com-

pared between the raw and denoised data. For the absorption coefficient, it is the profile of

absorption coefficient against frequency which distinguishes materials, and so the profiles

need to be compared. The raw and denoised data sets were compared pairwise in three

ways, using Pearson correlation, root mean square difference, and Student’s paired t-test.

4.3 Results

4.3.1 Blocked scans

Figure 4.4 shows a histogram and normal probability plot for the blocked terahertz signal

with a time constant of 500ms. The straight line indicates the hypothetical line on which

all the points would lie if the data were normally distributed.

The data shown have a mean of 4�65 � 10 � 4 and a standard deviation of 3

�72 � 10 � 5.

Plotting the data as a normal probability plot gives a PPCC of 0.999, which is larger than

the critical value of 0.9961 for 525 samples from a Gaussian distribution, which means the

null hypothesis can be accepted that the data are from a Gaussian distribution. However,

at the tail ends of the distribution there is the suggestion of a deviation from normality,

particularly at the bottom left.

Figure 4.5 is a plot of the same properties as Figure 4.4, but with a time constant

of 50ms instead of 500ms. In this instance the data shown have a mean of 4�64 � 10 � 4

47


Figure 4.4: A blocked terahertz scan (noise only) with a time constant of 500ms shownas (a) a histogram, and (b) a normal probability plot.

and a standard deviation of 6�25 � 10 � 5. Plotting the data as a normal probability plot

gives a PPCC of 0.9994 which is even larger than the critical value of 0.9961 for 525

samples from a Gaussian distribution, which means the null hypothesis can be accepted

in this case as well. Again a slight deviation from normality is suggested in the tails of

the distribution. Table 4.3 summarises these results.

Figure 4.5: A blocked terahertz scan (noise only) with a time constant of 50ms shown as(a) a histogram, and (b) a normal probability plot.

48


Time Constant Mean Standard Deviation Normal PPCC Accept?500ms 4

�65 � 10 � 4 3

�72 � 10 � 5 0.9990

�50ms 4

�64 � 10 � 4 6

�25 � 10 � 5 0.9994

�Table 4.3: Distribution of noise at different time constants

Time Constant α β γ δ Stable PPCC Accept?500ms 2.0 0.0093 0.00002 0.00047 0.9995

�50ms 2.0 -0.0561 0.00004 0.00047 0.9998

�Table 4.4: Fit of stable distribution to noise at different time constants

The simple model of a Gaussian distribution correctly models these data, so there is

no need to apply the GMM or stable distribution. However, for completeness a stable

distribution was fitted to this noise, and gave the parameters shown in Table 4.4. Notice

that the α value of 2 confirms this is a Gaussian distributio

4.3.2 Free air scans

The process from Section 4.3.1 was repeated with a set of 16 terahertz pulses transmitted

through free air. All 16 were recorded within a short space of time and with identical

acquisition parameters. In each case 256 time points were recorded, meaning the noise

distribution will consist of 4096 samples. There was a very small deviation between each

of the pulses, as demonstrated in Figure 4.6, which shows (a) the 16 pulses superimposed

on top of one another, and (b) the root mean square difference in the time domain between

the 16 pulses and the mean of the 16 pulses.

Figure 4.6(a) particularly demonstrates that the apparently noisy oscillations after the

main pulse are not in fact random noise at all — the 16 pulses are virtually indistinguish-

able. There is also the strong suggestion of a correlation between the amplitude and the

noise (as measured by the RMS difference). Figure 4.7 does not show that there is any

direct correlation between either electric field (a) or the absolute value of the electric field

49


Figure 4.6: (a) Superimposed time domain of 16 pulses through free air acquired withidentical parameters, and (b) the root mean square difference between the pulse and themean pulse at each time point.

Figure 4.7: The RMS difference (noise) versus (a) electric field, (b) absolute value ofthe electric field, and (c) the historical sum of the absolute values of the electric fieldincluding the previous 2 values, making a total of 3 values included in the sum (HAS3).Plot (d) is rescaled plot of (c), showing the bottom left hand corner in more detail.

(b). However, if the absolute value of the electric field is summed over 3 values ‘his-

torically’ there is a suggestion of a correlation. The sum was over 3 values because this

was determined empirically to give the greatest correlation, with rapid dropping off of

correlation with larger and smaller values.

50


Figure 4.8: Histogram of noise from a free air scan

Gaussian µ j σ 2j w j

G1 � 1�23 � 10 � 6 2

�12 � 10 � 8 0.79

G2 4�6 � 10 � 6 3

�52 � 10 � 7 0.21

Table 4.5: Gaussian mixture model created using EM for a free air scan.

Figure 4.8 shows the histogram of the estimated noise distribution.

The noise from these pulses had a mean of � 7�73 � 10 � 21 and a variance of 9

�11 �

10 � 8. The normal probability plot had a normal PPCC of 0.9179, far below the critical

value of over 0.9979 for 4096 samples, meaning this noise is not normally distributed.

Applying a GMM to the distribution gave rise to two Gaussians with the parameters shown

in Table 4.5. This GMM gave a correlation coefficient of 0.9882, a much better match,

but still below the critical value.

Figure 4.9(a) shows the normal probability plot, and (b) shows the GMM probability

plot. Notice that the noise clearly diverges from the hypothetical models at the tail ends

of the distribution, suggesting that the noise is heavy tailed.

Fitting a stable distribution resulted in a model with the parameters and PPCC shown

in Table 4.6. Figure 4.10 shows the probability plot from the stable distribution. The

PPCC value of 0.9999 is above the critical value, and the probability plot confirms the

goodness of fit.

51


Figure 4.9: (a) Normal probability plot, and (b) GMM probability plot from a free airscan.

α β γ δ PPCC accept?1.404 0.008 0.00012 � 0

�9 � 10 � 6 0.9999

�Table 4.6: Results for a stable distribution fitted to noise from a free air scan.

Figure 4.10: Stable distribution probability plot from a free air scan.

4.3.3 Mean pulse of step wedges

Table 4.7 shows the depth of each step of the wedges, and the number of pulses obtained

from that step using the region of interest technique, with each pulse having 128 time

points.

The average pulse for each particular step provided the estimate of a low-noise pulse

through that depth of material. The differences between the pulses through a given step

52


Nylon Count Size of Resin Count Size ofdistribution distribution

0.3 mm 32 4096 0.1 mm 43 55041 mm 53 6784 0.5 mm 95 121602 mm 40 5120 1.5 mm 59 75524 mm 44 5632 3 mm 49 62726 mm 31 3968 5 mm 79 101127 mm 13 1664 7 mm 60 7680

Table 4.7: Depth profiles and pulse counts of the step wedges.

and that step’s average pulse provided the estimate of the noise distribution for that depth

of material. These noise distributions were then assessed as before.

Figure 4.11 shows example normal probability plots for the thickest and thinnest nylon

steps. Table 4.8 shows the mean, variance and correlation coefficient obtained for each of

the 12 steps.

Figure 4.11: Normal probability plot of noise from a step of nylon of (a) 0.3 mm, and (b)7 mm.

These results clearly show that one cannot have confidence in using a single Gaussian

distribution to model this noise, as the critical value for these sizes of distribution is in

excess of 0.9979. The only exception to this rule is for the 7 mm thickness of resin, where

our critical value table does not extend to 7680 samples. However since it is only just

above the critical value for 1000 samples, it seems very unlikely to be above it at 7680.

Using the Gaussian mixture model, a far better fitting model was built, as shown in

53


Step Mean Variance PPCC Accept?Nylon 0.3 mm 0 399.83 0.7415 �Nylon 1 mm 0 209.04 0.758 �Nylon 2 mm 0 64.35 0.8491 �Nylon 4 mm 0 17.08 0.9371 �Nylon 6 mm 0 6.68 0.9912 �Nylon 7 mm 0 6.90 0.9799 �

Resin 0.1 mm 0 2873.53 0.7238 �Resin 0.5 mm 0 619.03 0.7382 �Resin 1.5 mm 0 113.86 0.7786 �Resin 3 mm 0 23.24 0.9922 �Resin 5 mm 0 21.93 0.9978 �Resin 7 mm 0 20.50 0.9985 ?

Table 4.8: Normal probability plot correlation coefficient for nylon and resin step wedges.

Table 4.9. Figure 4.12 shows an example GMM probability plot for the thinnest nylon

step, for comparison with Figure 4.11(a). Most of these correlation coefficient values are

smaller then the critical value for this number of samples, which means it cannot be said

that the noise distribution has been accurately modelled. In the cases where the critical

value is above the largest value I have, the probability plot still showed signs of heavy

tailedness. The EM algorithm was run with two and with three Gaussians, and it was

found that the weighting on the third Gaussian became infinitesimal very rapidly, meaning

that it was not contributing to the model. Only results for a mixture of two Gaussians are

therefore shown.

Finally a stable distribution was fitted to this noise data, resulting in PPCC values that

were well above the largest critical value I have. Since our critical values do not extend to

this size of distribution, one cannot be certain that the stable distribution is modelling it.

However the probably plots show an almost perfect line, similar to Figure 4.10. Table 4.10

shows the parameters fitted and the corresponding PPCC values.

These values were then plotted against step depth, as shown in Figure 4.13. For ease

of comparison, the α value of the blocked and free air scans are shown as dotted lines in

Figure 4.13. Note that the thickness axis in these two cases is meaningless. The γ values

54


Step Gaussian µ j σ 2j w j PPCC Accept

Nylon 0.3 mm G1 -0.0873 30.7 0.87 –G2 0.593 2905 0.13 –

∑w jG j – – – 0.960251 �Nylon 1 mm G1 -0.0986 13.8 0.86 –

G2 0.599 1395 0.14 –∑w jG j – – – 0.975396 �

Nylon 2 mm G1 -0.100 9.94 0.86 –G2 0.565 411 0.14 –

∑w jG j – – – 0.994245 �Nylon 4 mm G1 -0.132 5.24 0.81 –

G2 0.565 67.3 0.19 –∑w jG j – – – 0.994299 �

Nylon 6 mm G1 -0.0101 4.36 0.83 –G2 0.049 18 0.17 –

∑w jG j – – – 0.998484 ?Nylon 7 mm G1 0.00357 3.33 0.80 –

G2 -0.0141 21 0.20 –∑w jG j – – – 0.960251 �

Resin 0.1 mm G1 -0.152 259 0.90 –G2 1.434 27512 0.10 –

∑w jG j – – – 0.966960 �Resin 0.5 mm G1 -0.149 64.1 0.90 –

G2 1.29 5425 0.10 –∑w jG j – – – 0.960667 �

Resin 1.5 mm G1 0.0593 24.5 0.94 –G2 -0.992 1607 0.06 –

∑w jG j – – – 0.975454 �Resin 3 mm G1 0.0591 16.1 0.86 –

G2 -0.37 67.8 0.14 –∑w jG j – – – 0.999650 ?

Resin 5 mm G1 0.0367 18.3 0.90 –G2 -0.332 54.6 0.10 –

∑w jG j – – – 0.998484 ?Resin 7 mm G1 -0.0557 13 0.59 –

G2 0.0787 31.1 0.41 –∑w jG j – – – 0.999857 ?

Table 4.9: Gaussian mixture models fitted to the noise distribution from nylon and resinstep wedges. Numbers are shown to three significant figures, except the correlation coef-ficients.

55


Figure 4.12: GMM probability plot of noise from a step of nylon of 0.3 mm

Step α β γ δ PPCCNylon 0.3 mm 1.2167 -0.0055 3.8476 -0.062 0.99996Nylon 1 mm 1.1784 -0.0795 2.6364 0.057 0.99996Nylon 2 mm 1.3461 -0.01 2.3656 -0.05 0.99993Nylon 4 mm 1.4874 0.101 1.8335 -0.135 0.99982Nylon 6 mm 1.8571 0.0188 1.6535 -0.0078 0.99994Nylon 7 mm 1.6862 0.075 1.4902 0.00215 0.99988

Resin 0.1 mm 1.2954 0.013 10.8478 -0.137 0.99998Resin 0.5 mm 1.3366 -0.0065 5.5739 -0.111 0.99998Resin 1.5 mm 1.5453 -0.0055 3.3658 0.044 0.99997Resin 3 mm 1.8611 -0.0567 3.0958 0.0504 0.99992Resin 5 mm 1.9267 0.0613 3.1259 0.0273 0.99997Resin 7 mm 1.9407 0.0582 3.1088 -0.014 0.99989

Table 4.10: Stable distribution models fitted to the noise from nylon and resin stepwedges.

for the blocked and free air scans are too small to be seen on the scale of Figure 4.13(c)

— they are effectively both zero.

56


Figure 4.13: Stable distribution parameters against step thickness for nylon and resin stepwedges: (a) α , (b) β , (c) γ , (d) δ . Values for the blocked and free air scans are shown forcomparison.

4.3.4 Denoising

It was found that wavelet denoising has a negligible impact on the refractive index cal-

culations, even down to keeping only 10% of coefficients. Figures 4.14(a)–(e) show the

effect that wavelet shrinkage has on the pulse in the time domain. Notice that the main

peak is unaffected, and by shrinkages to 10% nearly all the low-level oscillations have

been removed.

Table 4.11 shows the time delays for each step at the various shrinkage levels, together

57


Figure 4.14: The time series samples of terahertz pulses with spectral content inset, atshrinkage to (a) 100 % (raw), (b) 50%, (c) 20%, (d) 10%, and (e) 5% of DWT coefficients.

with the final calculated refractive index. As can be seen, the extreme shrinkage at 5%

does cause differences in the time-delay calculations, but even then only a small error is

introduced to the refractive index value.

Similarly it was found that the absorption coefficient profile remained essentially un-

altered with only 20% of the wavelet coefficients. Table 4.12 and Figure 4.15 show the

absorption coefficient versus frequency profile at different shrinkage levels. On the plot,

the absorption coefficient values have been vertically offset for ease of view. By 10% the

58


Step Time delay (ps) at shrinkage level100% 50% 20% 10% 5%

Nylon 0.3 mm 0�49 0

�49 0

�49 0

�49 0

�492

0�02

20�02

20�02

20�02

20�03

Nylon 1 mm 1�91 1

�91 1

�91 1

�91 1

�932

0�02

20�02

20�02

20�02

20�02

Nylon 2 mm 3�94 3

�94 3

�94 3

�96 3

�9912

0�01

20�01

20�02

20�02

20�01

Nylon 4 mm 7�96 7

�96 7

�97 7

�97 7

�972

0�02

20�02

20�02

20�03

20�02

Nylon 6 mm 11�95 11

�95 11

�95 11

�92 11

�892

0�02

20�02

20�02

20�03

20�01

Nylon 7 mm 13�98 13

�98 13

�99 13

�99 14

�152

0�03

20�03

20�02

20�04

20�01

Refractive 1�603 1

�603 1

�604 1

�603 1

�61

index2

0�002

20�002

20�003

20�005

20�01

Table 4.11: Time delay and refractive index for different amounts of wavelet shrinkage.

(a)

(b)

(c)

(d)

(e)

Figure 4.15: Absorption coefficient profile vs. frequency for a nylon step wedge at shrink-age to (a) 100% (raw), (b) 50%, (c) 20%, (d) 10%, and (e) 5%, offset vertically.

59


feature at 1.1 THz has been lost, and with only 5% of coefficients large errors have been

introduced above 1 THz. The difference measures at the bottom are the Pearson correla-

tion, the root mean square difference, and the Student’s paired t-test probability. A t-test

suggests that the two series have no statistically significant difference at the 95% level,

even with the shrinkage to 5% of coefficients (Table 4.12). To ensure that important fea-

tures in the shape of the absorption coefficient curve, including the 1.1 THz feature, are

not removed, it would be prudent not to use the highest levels of shrinkage. A ‘safe’ level

of shrinkage for this experiment would therefore be 20%.

Frequency Absorption coefficient (cm � 1) at shrinkage level(THz) 100% 50% 20% 10% 5%0.52 4.63 4.61 4.43 4.29 4.540.57 5.14 5.15 5.30 5.50 5.030.63 6.68 6.68 6.66 6.87 6.130.68 7.29 7.29 7.26 7.75 7.910.73 8.27 8.27 8.30 8.41 8.640.78 9.22 9.21 9.23 9.16 10.130.83 9.87 9.89 10.02 10.24 12.410.89 11.00 10.98 11.01 12.14 12.870.94 12.35 12.40 12.80 13.51 13.800.99 14.33 14.34 13.71 14.23 15.431.04 12.94 12.91 13.46 14.33 18.531.09 16.27 16.28 16.20 14.94 21.171.15 19.56 19.37 18.45 18.11 14.821.20 21.76 21.71 21.88 19.91 13.351.25 21.69 21.55 21.41 21.50 12.561.30 22.14 22.35 23.94 22.96 12.091.35 24.28 24.39 24.22 24.72 11.901.41 26.54 26.35 25.33 26.98 12.171.46 28.29 28.19 28.13 30.33 12.91Pearson Correlation 1.0 0.9999 0.9966 0.9918 0.495R.M.S.D 0 0.09 0.61 0.97 7.05P-value frompaired t-test – 0.43 0.85 0.40 0.06

Table 4.12: Absorption coefficient (cm � 1) of nylon at various frequencies and shrinkagelevels, with measures of difference between each shrinkage level and the raw data.

60


4.4 Discussion

The first observation to make is that the electric field values recorded in the terahertz

pulses are arbitrary values, and so a direct comparison between the distributions obtained

for different acquisitions is not possible.

The second observation is that, unlike what might have been naıvely expected, the

noise distributions are not zero-mean. However, it is known that the terahertz scanner can

introduce a d.c. offset to the data, and this would account for the small shift away from

zero.

From the results of the blocked scan in Section 4.3.1 one can confidently say that

the background and detector noise in terahertz imaging may be modelled as a Gaussian

distribution, and altering the LIA time constant does not really change the mean of the

noise but does affect the variance — a shorter time-constant leads to a larger variance.

This is as expected, since the time constant is in effect a measure of how much averaging

the hardware is applying. Thus shorter time constants, or less averaging, lead to larger

variance in the noise distribution, or more noise. However with the blocked scan there

is a suggestion of ‘heavy tailedness’, as shown in the probability plot (Figure 4.5). The

points creep below the hypothetical straight line at the bottom left, and above it at the top

right, and so the apparent good fit may be due to under-sampling from the distribution.

The free air scan results demonstrate that the oscillations after the main pulse were

not random noise but rather were created by a deterministic process, probably absorption

and re-emission by water vapour [49], although another possibility is reflection inside

the crystals. However terahertz pulses through homogeneous regions of material showed

considerably larger random variation in these oscillations, perhaps due to scattering, sug-

gesting the working signal to noise ratio may actually be smaller. In any case, the random

noise in all these signals is not distributed in either a Gaussian distribution or a simple

combination of Gaussian distributions. There is also a relationship clearly visible be-

tween the arrival of the pulse and the noise level in Figure 4.6, even without a particular

61


correlation. It is possible that the arrival of the pulse is causing a ‘ringing’ effect as there

does appear to be some oscillation in the noise amplitude after the pulse. This is proba-

bly due to a physical effect such as reflection, or absorption and re-emission, but would

require further investigation.

The examination of a homogeneous region did not lead to a Gaussian distribution, but

did lead to a distribution which looked like a normal distribution with ‘heavy tails’ (Sec-

tion 4.3.3). The application of a Gaussian Mixture Model to the heavy tailed distribution

led to a far better match with the averaged region of interest data, although one still could

not be confident this was the correct model based on the probability plot. If the Johnson

noise forms a normal distribution, and the shot-noise is from a different normal distribu-

tion, then the overall noise will be modelled by a mixture of normal distributions, which

might explain the improved fit.

A stable distribution modelled the noise almost perfectly in all of these cases. One

would expect the noise distribution to be symmetrical, and hence the skewness parameter

β to be zero. Instead of this, β fluctuated around 0, depending on which material, if

any, had been imaged. β can take values from -1 to 1, and the largest β value fitted

was 0�1 for 4 mm of nylon. It is possible that these fluctuations may be related to the

scattering properties of nylon and resin, but this is impossible to tell without the inclusion

of a scattering model. As previously discussed, one would also expect the noise process

to zero mean, and hence for the location parameter δ to be zero. As before, the small

deviations from zero are probably due to d.c. offset errors in the lock-in amplifier.

Figure 4.13 suggests that α and γ might vary with thickness — as the material thick-

ness increases, α is tending to increase, and γ initially decreases, before flattening out.

Neither β nor δ particularly demonstrate a relationship with thickness, but rather both

fluctuate slightly around zero. Once again there may be a scattering process influencing

these variations.

The results from applying wavelet shrinkage show that only 20% of the wavelet co-

62


efficients are actually required when calculating optical properties. This suggests that a

wavelet shrinkage may actually be useful in a compression scheme, where only around

20% of the data need be transmitted and stored, which would represent a significant re-

duction in storage and transmission costs of data.

4.5 Conclusions

I have shown that the noise present in terahertz images can be modelled using distributions

from the stable family, and have calculated the parameters needed to model the noise for a

variety of terahertz images. I have also shown that, with the exception of the case when all

terahertz radiation is blocked, the noise present in terahertz images can not be modelled

by either a single or multiple Gaussian model.

Terahertz imaging is resilient to ‘classical’ noise such as shot noise, Johnson noise,

and background radiation due to the coherent nature of the detection. However inevitable

differences between and within samples, combined with other factors such as atmospheric

conditions, lead to a large variation in the pulses. The distribution of noise in terahertz

pulsed imaging is therefore more complex than was originally hypothesised. It had been

hoped that a computationally simple noise model, such as a single or dual Gaussian distri-

bution, would result from this analysis. The probability plots show that this is not the case,

but the noise can accurately be modelled by other distributions from the stable family.

Furthermore, the noise model created demonstrates dependence on both material type

and thickness, making it very difficult to build realistic models of noise, particularly for

material which has not been well characterised under terahertz. This in turn makes the

addition of entirely realistic noise to synthetic images an impossible goal until these noise

processes are better understood.

The apparent dependence on thickness also makes the design of algorithms that are

robust to terahertz noise more complex, as the thickness is often not known. Similarly

63


the lack of closed forms for the probability density functions of most stable distributions

make analytical methods impossible. However the distributions are essentially heavy

tailed Gaussians. In practice this means that there are more data points at the edges of the

distribution than would be expected from a Gaussian distribution, and it may be possible

to heuristically allow for these ‘extra’ points.

It is worth considering that these parameters may provide some form of insight into

the nature of the material being imaged. This may prove beneficial if a new material

produces similar results to a material previously characterised in this way.

It has also been shown that the wavelet shrinkage algorithm can use as few as 20%

of the wavelet coefficients from a step wedge image without introducing any significant

errors in the calculation of the optical properties. This result is of interest for medical ter-

ahertz pulse imaging since it could enable a real reduction in the storage and transmission

costs of scans, while not causing any change to results based on the physical properties of

the subject.

64

Chapter 5

Terahertz Imaging: Using Optical

Parameters as a Contrast Mechanism

5.1 Introduction

One of the most interesting aspects of terahertz pulsed imaging is the richness of data

acquired by virtue of the coherent detection in the time-domain. Typically one would

expect 512 or more samples to be acquired in the time-domain for each pixel in the image.

In this way a high dimensional vector is stored ‘behind’ every pixel of the ‘image’, rather

than just a single scalar value. Furthermore since the detection is coherent each pulse

may be transformed to the frequency domain. Combining the data from these domains

enables even higher dimensional vectors to be formed behind each pixel — just using

the time domain data, and the amplitude and angle of Fourier coefficients gives a 1536

dimensional vector for each pixel.

Not every dimension of these vectors contain useful information – for example the

signal to noise ratio at higher frequencies is too low for meaningful data to be extracted.

65

Chapter 5 Terahertz Imaging: Using Optical Parameters as a Contrast Mechanism

However there is scope for selecting smaller feature vectors or extracting other parameters

in order to create images [34, 46]. The data acquisition process can also be quite slow,

and although advances in the technology are expected to continue speeding up the imaging

process, it is worth exploring whether an entire time series need be recorded, or only a

subset thereof. Figures 5.1–5.4 show examples of simple parametric images. In each of

these figures a parameter has been extracted and mapped to grey level in order to form an

image.

(a) (b)

(c)

Figure 5.1: A cross section of a bird’s skull imaged in Frankfurt. 32 time points wereacquired at intervals of 150 fs, displacement -85. 60 � 48 pixels were acquired, with xrange -4 – 26 mm and y range -12 – 12 mm, and no averaging. (a) Reference peak tosignal peak time difference, white indicates largest delay. (b) & (c) Relative transmissionat (b) 1 THz and (c) 0.41 THz, white indicates largest transmission.

The differences in grey level at each pixel in these images is caused by two factors —

the thickness of material, and its optical parameters. The bird’s head is a slice of uniform

thickness, meaning the variation is entirely due to varying optical parameters. The nylon

and resin step wedges are of homogeneous material, meaning the variation is entirely due

to varying thickness, and the marzipan pig is somewhere between the two, with a mostly

homogeneous material having varying thickness.

66


(a) (b)

(c)

Figure 5.2: A nylon step wedge imaged in Frankfurt. The step wedges are fully describedon page 40, and the thinnest step is to the right of the image. (a) Reference peak to signalpeak time difference, white indicates largest delay. (b) & (c) Relative transmission at (b)1 THz and (c) 0.31 THz, white indicates largest transmission.

(a) (b)

(c)

Figure 5.3: A resin step wedge imaged in Frankfurt. The step wedges are fully describedon page 40, and the thinnest step is to the right of the image. (a) Reference peak to signalpeak time difference, white indicates largest delay. (b) & (c) Relative transmission at (b)1 THz and (c) 0.31 THz, white indicates largest transmission.

67


(a) (b)

(c) (d)

Figure 5.4: The head of a marzipan pig imaged in Frankfurt. 128 time points were ac-quired at intervals of 300 fs, displacement -77. 20 � 20 pixels were acquired, with x range-3 – 25 mm and y range -15 – 15 mm, and no averaging. (a) Photograph of the pig. (b)Reference peak to signal peak time difference, white indicates largest delay. (c) & (d)Relative transmission at (c) 1 THz and (d) 0.31 THz, white indicates largest transmission.

68


5.1.1 Complex Refractive Index

The complex refractive index, n, of a material gives the values of the two optical con-

stants n (refractive index), and k (extinction coefficient) according to (5.1), where both

parameters vary with wavelength λ .

n � λ �� n � λ �3 ik � λ � (5.1)

Together these constants describe the propagation of a given frequency of electromag-

netic radiation through a material, and conversely the complex refractive index may be

derived by observing the effect that a known thickness of material has on electromagnetic

radiation of a known frequency.

These parameters n and k are defined below, and are considered separately for the rest

of this study. Note that this thesis will also be using the absorption coefficient, α , instead

of k, where (5.2) describes the conversion between α and k.

α � 4πkλ

(5.2)

The optical constants are traditionally expressed in physics in terms of wavelength, as

in (5.1) and (5.2). In the context of this thesis, however, frequency (ω) is a more natural

expression and will be used for the rest of this chapter. The conversion between frequency

and wavelength is trivial, as shown in (5.3), where c is the speed of light in a vacuum.

ω � cλ

(5.3)

Refractive Index

The refractive index, n, for any substance is the ratio of the velocity of electromagnetic

radiation of a given frequency in a vacuum to its velocity in the substance. The relation-

ship between the phase angle, φ , of electromagnetic radiation of frequency ω transmitted

69


through a thickness x of material and the incident electromagnetic radiation, φ0, is given

in (5.4) [63].

φ � ω �� φ0 � ω �4 2πω � n � ω �5� 1 �c

x (5.4)

The refractive index provides a mechanism for contrast as radiation transmitted through

the same thickness of different materials will experience differing phase shifts.

Absorption Coefficient

The absorption coefficient is a measure of how much electromagnetic radiation of a given

frequency is absorbed by a given thickness of material. The Beer-Lambert law, shown

in (5.5), defines the relationship between the absorption coefficient α � ω � , the depth of

material x, and the relative transmission II0

(where I0 is the incident radiation and I is the

transmitted radiation).

ln � I � ω �I0 � ω � � �6� α � ω � x (5.5)

The absorption coefficient provides a mechanism for contrast as radiation transmitted

through the same thickness of different material will experience different attenuation.

This contrast is illustrated for two different frequencies in Figures 5.1(b) and (c).

5.1.2 Broadband Optical Properties

If a broadband signal, such as a pulse, is used, both the refractive index and absorption

coefficient have analogous pulse specific broadband versions, calculated from the time se-

ries data. The contrast in Figures 5.1(a), 5.2(a), 5.3(a), and 5.4(b) arises from differences

in the broadband refractive index. (5.6) shows the calculation of this broadband n using

peak to peak time difference (T D).

70


T D � � n � 1 � xc

(5.6)

(5.7) shows the calculation of this broadband α , I0 and I being incident and transmitted

peak amplitudes in the time domain.

ln � II0 � �6� αx (5.7)

5.1.3 Summary

As highlighted in Chapter 2, terahertz radiation is a novel imaging field, and very few

materials have been characterised under these frequencies of radiation. Consequently

there are no predications about the expected outcome of these analyses. It is hoped that

these data will prove useful for modelling the expected appearances of images and the

feasibility of future investigations.

In this chapter I present the results of calculating the refractive indices and absorption

coefficients of known thicknesses of material. Analogues to these parameters are calcu-

lated using broadband techniques, the short time Fourier transform (STFT) and the wide

band cross ambiguity function (WBCAF).

I also use the STFT with a rectangular window to simulate shorter acquisition times

and measure the effect that this has on the calculation of the complex refractive index.

5.2 Method

5.2.1 Data

In addition to the nylon and resin step wedges detailed on page 40, a number of biolog-

ical samples were imaged at Leeds. These consist of two sets of slices taken from two

amputated legs, with local Research Ethical Committee approval. Six different types of

71


tissue were excised; skin, adipose tissue (‘fat’), striated muscle (‘muscle’), vein, artery,

and nerve. Each tissue type was sliced into four different thicknesses; 50 µm, 100 µm,

150 µm, and 200 µm. Three pulses were acquired through each of the four thicknesses

of tissue, leading to twelve pulses in total for each tissue type per set. The first set of

slices was imaged twice over two days, having been stored in saline solution in a fridge

overnight.

Each pulse was a point measurement of 256 time points, acquired 80 fs apart, with a

displacement of 23.6. The time constant was 200 ms, with an LIA delay of 300 ms.

5.2.2 Traditional Analysis

The Fourier transform, introduced in Chapter 3, was used to calculate values of the de-

pendent variables for the calculation of the refractive indices and absorption coefficients

as a function of frequency. The techniques described here have been successfully applied

to terahertz data of water and other polar liquids by Kindt and Schmuttenmaer [41]. It

is also possible to calculate these parameters by fitting an expression that combines all

the reflection and propagation to the observed signal [18, 21], but these methods are not

robust.


The magnitude of each Fourier coefficient,�f � ω � � , from the transform of reference pulse,

gives values for I0 � ω � . Similarly the coefficients from the transform of a pulse transmitted

through a given thickness of material gives I � ω � for that thickness of material. The mean

of the logarithm of I � ω �I0 � ω � for each thickness of material can then plotted against thickness,

which should result in a straight line, the gradient of which is directly proportional to

α � ω � , as shown in (5.5). The error metric was the maximum variation in gradient per-

mitted within one standard deviation of the data points. In each case the gradient was

calculated using least squares linear regression.

72


Finally, it is necessary to apply inclusion criteria as at higher frequencies and thicker

steps the absorption becomes such that no meaningful terahertz signal passes through the

material. At any given thickness and frequency only values of I � ω �I0 � ω � greater than 0.05 were

used. This corresponds to a relative transmission greater than 5%. Additionally, α � ω � was

only calculated at frequencies where there were I � ω �I0 � ω � values for at least 3 thicknesses.

Refractive Index

The angle of each Fourier coefficient gives values for φ � ω � and φ0 � ω � for (5.4), enabling

essentially the same method to be applied for n � ω � as for α � ω � . The 5% relative trans-

mission results from the α � ω � were used to decide which pulses to include, and again

n � ω � was only calculated where there were at least 3 thicknesses.

There is extra processing required however, as the angles of the Fourier coefficients

are modulus 2π , i.e., range from � π to π , a feature of the sine and cosine basis for

this transform. These angles need to be ‘unwrapped’ to correctly reflect the phase data,

and this is carried out by assuming that any jump in angle between adjacent coefficients

greater than a threshold (usually π) is in fact ‘round the clock behaviour’. For example, if

φ � m � is � 3 and φ � m 1 � is 3, φ � m 1 � would be unwrapped to be 3 � 2π . Similarly

if φ � m 2 � was 2�5, this would be unwrapped to 2

�5 � 2π . Figure 5.5 shows an example

of the phase data before and after this process.

There are two sources of error in this process. Firstly there is an inherent ambiguity in

the unwrapping process — is a jump of exactly π due to modulus effects or not? Similarly

if there are genuinely jumps of more than π these will be removed. This effect can be

offset to some extent by padding out the time series with 0, which decreases the gaps

between the Fourier coefficients, hence making large ambiguous jumps less likely, and

then decimating the unwrapped coefficients back to the original length. This technique is

not a panacea however; not only does padding the time series increase the computational

load, but it is only of limited benefit in terms of reducing the errors in the n � ω � values.

73


Figure 5.5: (a) The raw angle of Fourier coefficient. (b) The result of unwrapping the an-gle of the Fourier coefficients for the reference pulse from (a) (red), and the three thinneststeps of nylon.

Indeed, after a certain level of padding it was found that the errors started increasing. In

this study the time series was padded until the errors in n � ω � were at a minimum. The

nylon and resin data were padded from 64 samples to 256 samples, and the biological

data were padded from 256 samples to 1024 samples.

The second source of error is cumulative error, whereby phase incorrectly unwrapped

in the first few coefficients carries down through the entire profile. This leads to phase

profiles for the same thickness of material proceeding down parallel paths that are mul-

tiples of 2π apart. The first few coefficients cannot be simply ignored as they establish

the relative positions of the profiles for different thicknesses of materials. To ignore them

would result in a change in separation between the profiles, which in turn would affect

the gradiant of the phase thickness plot, leading to an incorrect n � ω � value. This is partic-

ularly an issue with noisy data — the spectral content of the terahertz pulses tends to be

less for the first few coefficients, impacting the signal to noise ratio at these frequencies,

leading to noisy phase values. This error was minimised by skipping the noisy coeffi-

cients and then taking the mean of all the phase profiles for that thickness of material.

The profile closest to this mean was taken to be the correct interpretation, and the jumps

74


in the noisy section were adjusted to remove any 2π offsets. It was found that skipping

the first 10% of coefficients minimised the error in n � ω � .Broadband Analogues

The broadband pulse specific analogues of n � ω � and α � ω � , n and α respectively, were

calculated for comparative problems. In this case, (5.6) and (5.7) were used for the linear

regression plots. As these analogues are not frequency specific, there is just a single

value obtained for any given material. Once again the 5% relative transmission inclusion

criterion was applied, and every material gave values for at least 3 thicknesses.

These analogues are useful as a baseline comparison against which to compare the

alternative techniques. n is particularly useful given the unwrapping issues involved when

calculating n � ω � . They are also the properties that influence image contrast for images

based on parameters calculated in the time domain.

5.2.3 STFT

The short time Fourier transform, introduced in Chapter 3, was used to calculate n � ω � and

α � ω � . The STFT results in a 2D array of complex coefficients, β and ξ . The magnitude of

each coefficient represents the magnitude of signal present at that time (β ) and frequency

(ξ ). For every given ξ , the maximum magnitude of coefficient was used for calculating

α � ω � , and the β at which that coefficient occurred was used for calculating n � ω � . By

performing this operation on both the reference pulse and the sample pulses, comparable

values are obtained which may be used in the complex refractive index calculations.

The STFT requires two parameters to be set before analysis — the window function

and the width of that function. Three window functions were compared in this analysis —

a Gaussian function, a rectangular function, and a triangular function. These three win-

dow functions are ‘simple’ and well understood analytically, and, particularly in the case

of the last two, computationally ‘light’. However they do capture a range of properties

75


of windowing functions, from the smooth (Gaussian) to the sharply discontinuous (rect-

angular), and intermediate (triangular). The width of the Gaussian function was taken to

be the standard deviation of the distribution, whereas the width of the other two distri-

butions were taken to be the distance from the mid-point of the distribution to its edges,

that is where its value is 0. No normalisation was used with the rectangular or triangular

windows since relative values and ratios were used.

Finally, the short acquisition study was carried out by centering rectangular windows

of various widths on the peak of the pulse on the time-domain for each pulse in the dataset.

Following the windowing of the pulses, traditional frequency domain analysis was carried

out in order to plot the absorption coefficient profiles. In this way a collection of absorp-

tion coefficient profiles using various window widths was created. The window widths

were chosen from 1 sample up to 126 samples, in steps of 5, where the original pulses

consisted of 128 samples in total.

5.2.4 WBCAF

A refractive index analogue was calculated using the wide band cross ambiguity function

introduced in Chapter 3. The WBCAF was used to estimate the time-delay at scale 1, and

this estimate was then used in (5.6) to give the WBCAF absorption parameter. The time

delay was estimated by finding the translation τ which gave the maximum WBCAF value;

this corresponds to the amount the reference pulse must be shifted in order to achieve the

largest correlation, and hence best match, with the sample pulse. The WBCAF cannot be

used to estimate time-delay at different scales, since the scaling operation causes a shift

in the reference pulse.

The value of the WBCAF at any given scale and translation will be related to the

intensity of waveform with respect to the mother function, and this value was normalised

at a given scale σ using (5.8), giving an estimate of relative transmission at that scale.

76


MAXτ � WBCAFf1 f2 # τ σ "7�MAXτ 8 � WBCAFf1 f1 # τ 9� σ ":� (5.8)

This estimate of relative transmission was plotted against step thickness in order to

obtain an absorption parameter in a manner similar to the absorption coefficient calcu-

lations. In this way a profile of the WBCAF absorption parameter against scale can be

determined. This absorption parameter differs significantly from the absorption coeffi-

cient in that there is no exact correspondence between scale and frequency. Each σ will

correspond to a range of frequencies, and this range will vary with scale. This is unlike

the STFT, where there is an exact correspondence between ξ and frequency. A further

difference is that the band of frequencies represented varies inversely with σ - small σ

corresponds to high frequency content, and large σ corresponds to low frequency content.

77


5.3 Results

5.3.1 Traditional Analysis

Refractive Index

Calculating the refractive index analogue using the peak to peak difference in the time

domain as an estimate of time delay for (5.6) gives rise to a broadband refractive index, n.

Figure 5.6 shows the plot of peak to peak time delay against thickness for the resin step

wedge, where the gradient of the best fit line corresponds to the refractive index. The n

values calculated in this way are shown in Table 5.1.

Figure 5.6: Time difference against step depth for a resin step wedge.

Table 5.1: The broadband refractive index value, calculated in the time domain.

Material Refractive IndexNylon 1

�604

20�004

Resin 1�666

20�009

Skin 1�82

0�2

Fat 1�49

20�45

Muscle 2�12

0�4

Vein 1�60

20�77

Artery 2�75

20�64

Nerve 2�12

0�5

78


Finally, using the change of phase and (5.4) the profiles of n � ω � were created, and

these profiles are shown in Figure 5.7.

Figure 5.7: Refractive Index vs. frequency profiles for (a) nylon, (b) resin, (c) skin, (d)fat, (e) muscle, (f) vein, (g) artery, and (h) nerve.

79


In Figure 5.7 there are clearly problems with the biological samples. Although the

standard error on the linear regression is fairly small, the refractive index values do not

particularly correspond to the broadband values, and indeed in some of the values of n

are negative! These inconsistencies are generally due to the extremely noisy nature of

these data, and in particularly due to the difficulty of resolving the ambiguities when

unwrapping the phase.


In a similar way to that of the broadband refractive index calculation, the peak to peak

intensity ratio can be used as a measure of relative transmission in (5.7). However, this

gives rise to a broadband value α , which may not be linear with thickness and thus not be

described correctly by Beer’s law, as shown in Figure 5.8.

Figure 5.8: Broadband relative transmission against step depth for a resin step wedge.

The nonlinear relationship between the logarithm of broadband relative transmission

and step depth shown in Figure 5.8 renders the linear regression technique inappropriate,

and so this analogue was not calculated.

Figure 5.9 shows the logarithm of relative transmission against step thickness plotted

for a high frequency (1.5 THz) and a low frequency (0.5 THz), together with the pulse

height ratio plot from Figure 5.8 for ease of comparison.

80


Figure 5.9: The natural logarithms of relative transmissions at 0.5 THz and 1.5 THzagainst step thickness for a resin step wedge, with the broadband time domain basedvalues included for comparison.

Figure 5.9 clearly demonstrates that absorption changes with frequency, as different

gradients result for 0.5 THz and 1.5 THz. This suggests α � ω � varies across this frequency

range in this material, as onewould expect, and indeed α � ω � varies for all the samples in

this frequency range. Figure 5.9 also highlights the need for inclusion criteria in these

calculations - the thickest two resin steps no longer demonstrate Beer’s law at 1.5 THz,

since all the radiation of and above this frequency has effectively been absorbed.

The α � ω � of each material was calculated, and are plotted in Figure 5.10. Below

around 0.5 THz and above 1.5 THz the signal is swamped by noise, especially with the

thicker samples of material, and so the profile is only shown within these boundaries.

Results are reported for nylon for the slightly lower frequency band 0.13 to 0.48 THz by

Birch et al. [5], where the values for α � ω � are comparable with ours.

The large errors in the biological samples are believed to be the result of a combination

of factors, including the normal biological variation between the two donors, the hydration

state of the tissue varying as the imaging progressed, inevitable small differences in the

thickness of tissue presented, and scattering losses.

81


Figure 5.10: Absorption coefficient vs. frequency profiles for (a) nylon, (b) resin, (c) skin,(d) fat, (e) muscle, (f) vein, (g) artery, and (h) nerve.

82


5.3.2 STFT

Figure 5.11 shows the output from the STFT when it is applied to terahertz pulses. The

Gaussian window function was used with a width of 9 ps (60 samples) to demonstrate the

time/frequency nature of the STFT. Notice that the retardation of the pulse by the nylon is

visible in the STFT spectrogram, as the larger coefficients (denoted by darker grey level)

are further to the right than the reference pulse. The attenuation is also visible, as the

coefficients are all smaller (lighter grey levels). Finally pulse broadening can be seen, as

the widest part of the nylon pulse’s STFT coefficients is wider than those of the reference

pulse’s STFT.

0 5 10 15-1000

0

1000

2000

3000

Time (ps)

Ele

ctri

c F

ield

Fre

quen

cy (

TH

z)

0

0.5

1

1.5

2

2.5

3

(b)(a)

0 5 10 15-1000

0

1000

2000

3000

Time (ps)

Ele

ctri

c F

ield

0 5 10 15

Time (ps)

0 5 10 15

Time (ps)

Fre

quen

cy (

TH

z)

0

0.5

1

1.5

2

2.5

3

Figure 5.11: Time domain of terahertz pulse and corresponding STFT coefficients for (a)the reference pulse, and (b) a pulse through 2 mm of nylon. The grey level representsthe mutually normalised magnitude of the coefficient, with black indicating 1 and whiteindicating 0.

83


Refractive Index

Figure 5.12 shows three examples of the graph of step thickness against pulse delay for

different window width of the STFT. The STFT was performed with the Gaussian window

function, and the delays taken were for 1 THz. Note how the error bars increase and the

straight line fit is less good as window width increases. This is to be expected, as the

resolution balance changes from being temporally biased to being spectrally biased.

Figure 5.12: Time delay vs step thickness for a nylon step wedge at 1 THz. Calculatedusing the STFT with a Gaussian window of widths (a) 10 � 1 (0.015 ps) (b) 101 (1.5 ps) (c)103 � 5 samples (474 ps).

84


Figure 5.13 shows the refractive index profiles of the nylon step wedge, calculated

using the STFT with a Gaussian window, with widths from 10 � 1 to 103 � 5. These widths

correspond to around 0.015 ps and 474 ps respectively. The exponential changes are re-

quired to fully demonstrate the effect of changing the window width.

Figure 5.13: Refractive index profiles for a nylon step wedge using the Gaussian win-dowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps), (d)100 � 5 (0.47 ps), (e) 101 (1.5 ps), (f) 101 � 5 (4.7 ps), (g) 102 (15 ps), (h) 102 � 5 (47 ps), (i) 103

(150 ps), and (j) 103 � 5 (474 ps).

Figures 5.14 and 5.15 show the equivalent figures for the triangular and rectangular

windows respectively. Rectangular windows wider than 10 samples gave refractive index

results that were completely swamped with noise, and so the profile is only shown for

windows narrower than this.

85


Figure 5.14: Refractive index profiles for a nylon step wedge using the triangular win-dowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps), (d)100 � 5 (0.47 ps), (e) 101 (1.5 ps), (f) 101 � 5 (4.7 ps), (g) 102 (15 ps), (h) 102 � 5 (47 ps), (i) 103

(150 ps), and (j) 103 � 5 (474 ps).

86


Figure 5.15: Refractive index profiles for a nylon step wedge using the rectangular win-dowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps), (d)100 � 5 (0.47 ps), and (e) 101 (1.5 ps).

As expected, the narrowest window results in a completely flat refractive index pro-

file at the level of the broadband value, no matter which window shape is used. This

is because the narrow limit of the STFT is effectively calculation in the time domain,

with no frequency dependence. Of course, the Fourier based refractive index profile also

demonstrates little variation against frequency, and this is reflected in the plots for most

of the window widths. The triangular window function appears to be the most stable for

this calculation, whereas the Gaussian and rectangular functions both demonstrate large

fluctuations away from truth as the windows get wider. However, the Gaussian window

function does demonstrate the transition from temporal to spectral emphasis. How the

triangular window function introduces spectral artefacts has also been seen, and so only

the Gaussian window has been used for the rest of the samples. These results are shown

in Figure 5.16. The complete set of graphs showing how the refractive index profile varies

with window width for each sample type may be found in Appendix B.

All of these graphs show refractive indices comparable with the broadband values

calculated in the time domain. The skin, artery, and nerve analyses all give values that

87


Figure 5.16: Refractive index profiles calculated using the Gaussian windowed STFTwith width 101 � 5 (6.3 ps) of (a) a resin step wedge, (b) excised skin, (c) excised fat, (d)excised muscle, (e) excised vein, and (f) excised artery

are somewhat lower than the broadband value, although still within the bounds of error

from these calculations. Only the fat and vein refractive indices bear much resemblance to

the Fourier calculated refractive indices. The dissimilarity on the other samples is almost

certainly due to the problems in unwrapping the phase of these noisy biological data.

In every case, it seems to be windows of 10 samples or wider that give rise to refrac-

tive indices which are dependent on frequency, and it is generally true that within this

frequency band the refractive index seems to be almost constant.


The absorption coefficient at a given frequency is calculated from the gradient of the log-

arithm of the ratio of incident to transmitted intensities versus step thickness. Figure 5.17

shows three examples of such gradients, all using the STFT with the Gaussian window

function and calculated at 1 THz but with different widths of window function. By 1 THz,

the deepest nylon steps have less than 5% transmittance, which results in a very low signal

88


to noise ratio. To avoid this noise, only steps with transmittance greater than 5% are used.

The straight line shows the gradient extracted by this method — the solid part indicates

which steps thicknesses were used in calculating this gradient.

Figure 5.17: Logarithm of transmittance vs step thickness for a nylon step wedge at 1 THz.Calculated using the STFT with a Gaussian window of widths (a) 10 � 1 (0.015 ps) (b) 101

(1.5 ps) (c) 103 � 5 samples (474 ps). The points near the solid line are those that were usedin calculating the gradient, the others were excluded by the noise criteria.

As before, applying the STFT to the terahertz data enables us to calculate the absorp-

tion coefficient profiles for different window widths and window functions. Figures 5.18

to 5.20 show the profiles of the nylon step wedge for a Gaussian, triangular, and rectan-

gular window function respectively.

In the case of the absorption coefficients, the results become worse as the window gets

narrower. This is entirely expected, since the absorption is calculated from the intensities

of frequencies present. The cut-off point at which the 1 THz absorption feature is lost

89


when using windows narrower than 101 � 5 samples (4.7 ps) for the triangular or Gaussian

window functions, and 102 samples (15 ps) for the rectangular window function. There

is little to differentiate the three different window functions, except that the rectangular

window has a smaller range of widths within which its results are meaningful.

Figure 5.18: Absorption coefficient profiles for a nylon step wedge using the Gaussianwindowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps),(d) 100 � 5 (0.47 ps), (e) 101 (1.5 ps), (f) 101 � 5 (4.7 ps), (g) 102 (15 ps), (h) 102 � 5 (47 ps), (i)103 (150 ps), and (j) 103 � 5 (474 ps).

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Figure 5.19: Absorption coefficient profiles for a nylon step wedge using the triangularwindowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps),(d) 100 � 5 (0.47 ps), (e) 101 (1.5 ps), (f) 101 � 5 (4.7 ps), (g) 102 (15 ps), (h) 102 � 5 (47 ps), (i)103 (150 ps), and (j) 103 � 5 (474 ps).

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Figure 5.20: Absorption coefficient profiles for a nylon step wedge using the rectangularwindowed STFT with widths (a) 10 � 1 (0.015 ps), (b) 10 � 0 � 5 (0.047 ps), (c) 100 (0.15 ps),(d) 100 � 5 (0.47 ps), (e) 101 (1.5 ps), (f) 101 � 5 (4.7 ps), (g) 102 (15 ps), (h) 102 � 5 (47 ps), (i)103 (150 ps), and (j) 103 � 5 (474 ps).

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Figure 5.21 shows how the absorption coefficient varies against frequency with a

Gaussian window function of 101 � 5 samples. The complete set of graphs showing how

the absorption coefficient profile varies with window width for each sample type may

again be found in Appendix B.

Figure 5.21: Absorption coefficient profiles calculated using the Gaussian windowedSTFT with width 101 � 5 (6.3 ps) of (a) a resin step wedge, (b) excised skin, (c) excisedfat, (d) excised muscle, (e) excised vein, and (f) excised artery.

In these graphs the absorption coefficients mimic the refractive indices in that the

narrowest window functions give an unchanging absorption coefficient against frequency,

meaning that little spectral information is being used in the calculation. The absorption

coefficient profile obtained using windows of 101 � 5 samples and wider follow the basic

trend of the Fourier based profiles, although in each case there is clearly a smoothing

effect, as would be expected through the application of a Gaussian window.

These results together suggest that, for these acquisitions, an STFT analysis with a

Gaussian window function of width of around 31 or 32 samples strikes an appropriate

balance between spectral and temporal resolution, and thus enables a good approximation

of the refractive indices and absorption coefficients to be calculated.

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Short Acquisition Simulation

Figure 5.22 shows the absorption coefficient profiles of the nylon and resin step wedges

when the pulses have been windowed with a rectangular window function of various

widths. The widths are given in time rather than number of samples for ease of compari-

son between the two different scans, since the data were recorded with different sampling

rates. The widths chosen demonstrate the narrowest window that gives comparable results

to the unwindowed results, a medium width window, and a narrow window.

I found that a window width of at least 111 samples (16.5 ps) was required before all

the features of the nylon step absorption profile were captured. Similarly 106 samples

(21.20 ps) were required for the resin step wedge. The nylon step was originally imaged

using 128 samples, or 19.2 ps. The resin step also used 128 samples, but since there was

a larger gap between the sample points, this corresponds to 25.6 ps. Note these values are

not the time taken to acquire the pulses, since each individual sample point takes many

seconds to acquire. Instead, these figures refer to the amount of time it would take to

capture the pulse if it could be acquired in real time.

However, the overall trend of the absorption coefficient against frequency is retained

even with very small windows. Figure 5.22 demonstrates that the trend is broadly fol-

lowed for even a 2.4 ps window (16 samples). With the resin step wedge, a 1.2 ps window

(6 samples) results in an absorption coefficient profile that follows the true trend closely.

These correspond to a total acquisition of 32 and 12 data points respectively, compared

with 128 on the original pulses. This in turn corresponds to a reduction in acquisition

time to about 25% and 10% of the original acquisition time respectively. Once smaller

window functions than these were used, large errors were introduced in the α plots.

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Figure 5.22: The absorption coefficient profiles of fixed rectangular windowed data from(a)-(c) a nylon step wedge, and (d)-(f) a resin step wedge.

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5.3.3 WBCAF

Figure 5.23 shows the coefficients when the WBCAF applies the reference pulse to a

pulse through 2 mm of nylon. Notice that the coefficients show a drift to the left as the

scale increases down the figure — this is due to the increasing scale having the side effect

of shifting the pulse to the right.

Figure 5.23: The WBCAF coefficients of a pulse though 2 mm of nylon. White pixelsindicate the largest positive coefficients, black indicates the largest negative coefficients,and grey pixels in between, with most being almost zero.

5.3.4 Refractive Index

The refractive indices of nylon and resin were calculated using the translation of the max-

imum WBCAF parameter as an estimate of time delay, as described in Section 5.2.4.

Figures 5.24(a) and (b) show plots of step depth against estimated time delay, together

with the line of best fit, calculated using linear regression. Table 5.2 shows the values of

refractive index calculated with the WBCAF, together with the broadband values for com-

parison. As with the broadband calculations, there is a very good fit achieved by linear

regression, and this is reflected in the low error boundaries.

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(a) (b)

Figure 5.24: WBCAF time delay estimate against step-depth with best fit line for (a)nylon, and (b) resin, both at scale 1.

Nylon ResinTime Domain 1.603

20.004 1.66

20.01

WBCAF 1.5972

0.005 1.642

0.02

Table 5.2: Refractive indices calculated by traditional methods and using a WBCAF.

5.3.5 WBCAF Absorption Parameter

The absorption parameter was calculated using the normalised maximum WBCAF coef-

ficients from (5.8) to estimate relative transmission. Figure 5.25 shows the logarithm of

this relative transmission for the nylon and resin step wedges.

(a) (b)

Figure 5.25: Logarithm of the normalised WBCAF maxima at scale 1 for (a) nylon and(b) resin step wedges. Black indicates the smallest value, the most absorption, and whiteindicates the largest value, or least absorption.

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Figure 5.26 shows the plots of the WBCAF estimated relative transmission against

step depth for (a) resin, and (b) nylon, at scale 1. The points in the nylon plot approximate

a straight line, however the resin plot displays a similar curve to the time domain based

plot in Figure 5.8. This is to be expected given the larger absorption coefficient of resin

compared to nylon, and the range of frequencies present within each individual WBCAF

scale.

(a) (b)

Figure 5.26: Relative transmission, as estimated by the WBCAF, against step thicknessfor (a) nylon, and (b) resin.

Finally Figure 5.27 shows the profile of the WBCAF estimated absorption parameter

against scale for both nylon and resin. The absorption coefficient values decrease with

increasing scale as scale is inversely related to frequency content. This parameter clearly

distinguishes between these two materials, although the degree of difference in absorption

parameter is much larger then the difference in α � ω � .5.4 Discussion

As the figures of refractive indices and absorption coefficients show, these parameters

enable the different materials used in this work to be distinguished, at least at some fre-

quencies.

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Figure 5.27: WBCAF absorption parameter against scale for nylon and resin.

5.4.1 STFT

The absorption coefficient profiles generated using the STFT are entirely consistent with

the traditional analysis techniques. Naturally as the window function widens, the results

come closer and closer to the Fourier based analysis. Similarly the refractive index results

are consistent with the time-based values. As can be seen, the narrowest windows give

refractive index values which are almost independent of frequency, but which correspond

closely to the time domain calculated value. The STFT also enables the generation of

n � ω � vs. frequency that doesn’t depend on phase, and therefore avoids the difficulties

associated with that method.

The rectangular window function performs most poorly, giving meaningful refractive

index and absorption coefficient results with only a small range of window widths. There

is no overlap between the two results — a window narrow enough to give meaningful

refractive index results is too narrow to give absorption coefficient results, and a window

wide enough to give absorption coefficient results is too wide to give refractive index

results.

The right balance must be struck between time and frequency resolution. As has

been shown, narrow windows achieve good temporal resolution at the expense of spectral

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resolution, while the opposite is true for wide windows. The obvious question is how wide

(or narrow) a window must be in order to achieve meaningful results in both temporal and

spectral spaces.

Fourier analysis gives ‘perfect’ results for α � ω � , and so how α � ω � varies with fre-

quency is known. The α calculated by the STFT can therefore be evaluated against this

ground truth. In this way, the minimum window width of a given window function needed

to retain spectral resolution can be found. One can then be certain that using a window of

this width (or wider) will give meaningful results in terms of spectral resolution.

The problem is then to find the maximum window width that can be used that still

retains sufficient temporal resolution for the analysis of refractive index. We have no

‘perfect’ results for the refractive index against frequency, but if the refractive index values

are constant against frequency, we can be confident that not enough spectral information

is being used in that analysis, i.e., the window is too narrow. Similarly, large error bars and

large fluctuations in the refractive indices (especially if it ever goes negative) indicate that

not enough temporal information is being used, i.e., the window is too large. In this way

an estimate can be obtained of the maximum possible window width at which temporal

resolution is retained.

If the maximum window width usable to retain temporal resolution is smaller than the

minimum width usable to retain spectral resolution (that is, there is no overlap of window

widths), then the window function itself is unsuitable for joint time/frequency analysis.

The rectangular window function falls into this category, although neither the triangular

nor Gaussian functions do.

5.4.2 Simulation of short acquisition

The simulations of short acquisition results were disappointing; almost the entire pulse

was needed to capture all the features of the absorption coefficient profile. The width of

the rectangular window function refers to the distance from the centre of the window to its

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edge, that is when its value becomes zero. Thus the window is in reality twice the width

of its ‘width’ parameter. This means that if the window is centered about the middle of the

pulse then a window ‘width’ of 64 samples will capture the entire pulse for the nylon and

resin data. If the window is centered on the peak of the pulse then the amount of the pulse

that remains after the windowing will depend on the position of that peak. The thinnest

nylon step has its peak at 15 samples, meaning that the 111 width window will capture

the first 126 samples — effectively the entire pulse. The thinnest resin step has its peak at

17 samples, meaning that the 106 width window will capture the first 123 samples, just a

little short of the entire pulse. In both cases, the thickest steps had pulses which were at

103 and 94 samples for the nylon and resin respectively. Both of these values mean that

the entire pulse remains after windowing.

However, the overall trends of αω are captured, even when using very small windows

(short acquisition times). Thus it may save valuable time if only a false colour dual- or

tri-frequency image is being formed. An example of such an image might be colouring

the pixels such that the red component corresponds to α at 0.5 THz, and the blue to α at

1.3 THz (Loffler et al. demonstrate this sort of visualisation in [46]). In such a situation

the pulse would be rapidly scanned to locate the peak, and then the appropriate number

of samples either side of that peak acquired.

The reduction in acquisition time to 10 or even 25 percent of the original acquisition

time are also highly significant when it is considered that the original scans took over

eight hours each.

5.4.3 WBCAF

The WBCAF results for the refractive index are entirely consistent with the broadband

value, but the absorption parameters are harder to assess. Clearly the profiles for nylon

and resin are quite different, and so this parameter shows promise as a contrast mecha-

nism. The actual values are also the same order of magnitude as the Fourier based values.

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Scale is not directly comparable to frequency, although frequency does correspond to in-

verse scale in some manner. With decreasing scale, and hence increasing frequency, the

separation between the nylon and resin absorption increases, mainly due to the increase of

the resin’s absorption. This does not mirror exactly the true absorption coefficient values,

although the absorption coefficient of resin is generally larger than that of nylon. This

inconsistency with the true values for absorption coefficients does raise the obvious con-

cern about whether the WBCAF values are actually meaningful and a useful metric. This

is not a question that can be answered with the limited data currently available.

The WBCAF does also have a number of disadvantages over the traditional tech-

niques. Firstly the method is very computationally expensive, as the reference pulse has

to be interpolated and filtered for every scale in the analysis. Secondly, the effective shift-

ing of the reference pulse causes difficulty with calculating the phase changes or time

delays, and hence refractive indices. Finally the results are not wholly consistent with the

traditional analyses.

5.5 Conclusions

The high dimensionality of terahertz data is a double edged sword — while providing

a richness of data it also requires more sophisticated processing techniques than simple

scalar data. In this chapter I have explored some techniques for reducing this high dimen-

sionality through the calculation of the complex refractive index and its analogues.

The short time Fourier transform is the most promising of these, handling the noisy

biological data while still giving results consistent with traditional analyses. This is a

tool which might be especially useful in reflection mode imaging, when the ability to

determine the spectral content at a given point in time becomes crucial. Further analy-

sis needs to be carried out in order to determine whether the STFT analysis sufficiently

distinguishes between materials to actually be useful. Similarly in this study a Gaussian

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window was shown to be an appropriate choice of window function, and a rectangular

window was shown not to be.

The cross ambiguity function showed itself to be theoretically capable of distinguish-

ing between nylon and resin based on refractive index and an absorption parameter, how-

ever the absorption parameter did not mirror the absorption coefficient, suggesting that it

is not truly the absorption which has been calculated.

A joint time-frequency analysis has three distinct advantages over a traditional ap-

proach; it enables the refractive index to be unambiguously calculated as a function of

frequency, it enables ‘one-shot’ signal processing, where both refractive indices and ab-

sorption coefficients are calculated in one pass over the data, and, perhaps most impor-

tantly, it compromises spectral resolution in order to gain temporal resolution, so that not

only is the spectral content obtained, but also when that content occurred — a point that

is of special interest in reflection mode imaging.

Finally, the effects that shortening the acquisition time have on traditional calculations

of the optical properties were analysed. It was interesting to see that an extremely short

acquisition time still led to an absorption coefficient that captured the general trend of

the full-sample absorption coefficient. However, such short acquisition times do cause

spectral features to be lost, and future work may demonstrate that these features are key

in distinguishing between materials.

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Chapter 6

Approaches to Segmentation of

Terahertz Pulsed Imaging Data

6.1 Introduction

As discussed in Chapter 5, terahertz pulsed imaging delivers data of a high dimension-

ality. This high dimensionality raises the possibility of clustering techniques to segment

terahertz images into their constituent regions. Groups of pixels which are ‘similar’ in

some way are clustered together, and an image may then be formed by the assignment

of different colours or greyscale levels to these groups. These images may then either be

used as an end in themselves, or alternatively could be used as a first step that highlights

potential areas of interest for further analysis.

The key advantage that this approach has over analyses based on the optical properties

of the sample is that very little information about the sample is required. For instance, it

is hard to calculate the complex refractive index if the depth of material is not known, or

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Chapter 6 Approaches to Segmentation of Terahertz Pulsed Imaging Data

if several types of material are present in the sample. Clustering techniques, on the other

hand, bypass these problems by just working on the theory that ‘similar’ pixels are likely

to have similar constituency. Of course, a good similarity metric is vital if the clustering

is to be successful, and this chapter explores some of the options in the choice of metric.

In this chapter, I apply k-means clustering to a variety of terahertz data, based on both

high dimensional vectors formed from temporal, spectral, and combined domains, and

on low dimensional vectors formed by extracting specific features such as absorption at a

given frequency.

6.2 Method

6.2.1 Data

Synthetic Slice of Tooth

In order to assess the accuracy of the segmentations, a synthetic terahertz data set of a

slice of tooth was created. A model of a slice of tooth was taken, consisting of three

regions; air, enamel, and dentine, as shown in Figure 6.1(a). The number of pixels was

then decreased to reflect the spatial resolution of terahertz, resulting in an image of 50x50

pixels, consisting of 1585 pixels of air, 201 pixels of enamel, and 714 pixels of dentine

(Figure 6.1(b)). The boundary effects between the different tissue types were intentionally

ignored in this image, since this is a first stage exploration of the segmentation techniques.

A synthetic image built in this way leads to a perfect ground truth for the segmentation as

each pixel in the image belongs to exactly one class, which is already known. In this way,

each segmentation can be evaluated by counting the number of pixels which have been

incorrectly clustered.

To create the teraherta data set, a terahertz time-series must be ‘acquired’ for each

pixel — as if the model was an actual object that was being imaged. This was done by

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Figure 6.1: (a) An example cross-section of a tooth, showing the enamel and dentineareas, and (b) the allocation of regions in the synthetic data.

using 3 exemplar terahertz pulses from a real scan of a slice of tooth — one per region —

and using them to populate the terahertz data set as appropriate. Thus all the air regions

within the terahertz data set consist of an indentical terahertz pulse that was actually

acquired through air, and similarly for the dentine and enamel regions.

The disadvantage of a pure synthetic image is that segmentation becomes trivial — the

intra-region differences are zero while the inter-region differences are large. To attempt

to make a more realistic model random normally distributed noise was addedd to the time

series. This study was undertaken before the noise investigation of Chapter 4 and without

that knowledge, therefore a simple noise model was used.

Real Slice of Tooth

A 200 µm cross section of tooth sliced from a real tooth was imaged in transmission

mode at Leeds. 64 time points were acquired at 146.7 fs intervals, with no averaging and

an offset of 22.5. The time constant was set to 500 ms, with an LIA delay of 1 s. 56 scans

were obtained in each of the x and y directions, with x ranging from 0 to 22.2 mm and y

ranging from 0 to 9 mm.

An x-ray of the slice of tooth was taken, and was scaled and registered to a high

frequency terahertz phase image, using the external edge of the tooth and the hole as the

surfaces to be matched, in Analyze. Segmentation was done interactively in Analyze by a

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single observer to obtain the ground truth for comparison. Figure 6.2 shows the segmented

radiograph of the tooth slice and a terahertz absorption image.

(a) (b)

Figure 6.2: (a) A hand segmented radiograph of a real slice of tooth, and (b) the relativetransmission image of the real slice of tooth at 1.38 THz.

Phantoms

Finally, phantoms were created specifically for the purpose of developing and evaluating

the segmentation techniques. The desireable properties are for the phantoms to have dis-

tinct, easily quantifiable regions of different, well-characterised materials. Unfortunately

at the time of writing there were very few well characterised materials in the terahertz

band, and those that were did not lend themselves well to the fashioning of clustering

phantoms within the scope of this thesis.

It was decided to use a hollow (or outline) sticker whose regions could be filled with

different materials because this immediately provides a phantom that has distinct well

definied regions, that can also be easily segmented by hand from a visual image and

subsequently registered to its terahertz image. Each region defined by the sticker was

carefully painted with a domestic paint until it was completely filled. Various types and

colours of paint were chosen in the hope that these would provide enough contrast in

the terahertz band to make the segmentation non-trivial but still possible. Two further

107


phantoms were manufactured - one based solely on a sticker, and one based solely on

paint.

The phantoms were manufactured on TPX sample plates with four different domestic

paints1. The sample plates were approximately 25 mm square with 2 mm thickness of

TPX, and are usually used as the front and back of the sample holders.

The paints are identified by colour, and are detailed in Table 6.1. The stickers were

peel off ‘gold’ stickers obtained by mail order from kookykards.com 2, with the ‘rabbit’

coming from sheet number 1850 (“Farm Animals”) and the ‘foot’ coming from sheet

number 1844 (“Francoise Read: Washing line and Feet”). Table 6.2 details the construc-

tion of each of the three phantoms. Photographs of the phantoms are shown in Figure 6.3,

although note that the ‘THZ’ phantom is shown before the top coat was applied.

Colour Description NameBlue Homebase matt vinyl emulsion for interior walls and

ceilings“Fisherman’s Blue”

Red Homebase “Sanctuary” matt emulsion “Firedance”Gold Crown metallic effect “Gold Palace”Green Crown shimmer coat “Green Shimmer”

Table 6.1: Description of the paints used to create the phantoms.

Name DescriptionFoot A solid sticker of a foot (taken from inside the hollow outline).Rabbit A hollow sticker of a rabbit, having two large regions filled with red and green

paint, and two smaller regions filled with yellow and blue paint.THZ A single coat of blue paint, followed by the letters ‘T’, ‘H’, and ‘Z’ painted

in green, followed by a covering coat of blue paint.

Table 6.2: Description of the sticker and paint phantoms.

The sticker phantoms were designed to be imaged in transmission mode, however

operational issues beyond our control prevented the acquisition of these data at Leeds,

1TPX, or poly 4 methyl pentene-1, is a low loss polymer that is almost transparent at terahertz frequen-cies [5].

2http://www.kookykards.com/peel off stickers images.htm Last visited November 27, 2003.

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(a) (b) (c)

Figure 6.3: Photographs of the phantoms (a) ‘foot’, (b) ‘rabbit’, and (c) the partiallycompleted ‘THZ’ phantom.

and instead they were imaged in reflection mode. They were imaged at Teraview Ltd.,

Cambridge, with the acquisition parameters shown in Table 6.3. The top part of the table

lists the machine settings, in upper case, while the lower section lists some useful derived

parameters. Note that the x direction in the Teraview machine is scanned continuously

by sampling at constant time intervals as the x stage moves. In this way, approximately

constant spatial x intervals between the pixels is achieved. This means that the number

of pixels acquired in the x direction is dependent on the distance being scanned and the

speed of the x stage, according to the formula total pixels ; total x distance (mm)xspeed < scanfreq.

The scanning did vary however, and only 99 x pixels were obtained in the ‘rabbit’ image,

whereas 101 x pixels were obtained in the ‘THZ’ image. The ‘foot’ had exactly 100 x

pixels.

The pulses demonstrated a large d.c. offset, and this was removed using the fast

Fourier transform before any clustering was undertaken. The phantoms were imaged

face-up, with the bottom of the TPX sample plate resting on the imaging window. This

arrangement results in a time-series consisting of two pulse responses, as shown in Fig-

ure 6.4. The first of these is the response from the quartz/TPX interface, and in a perfect

system would be identical at every pixel. The second pulse is the response from the

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TVL Version 3DATASIZE 2048APEAMP 800.0PHASE 2SCANFREQ 15.0HIGHFILTER 16AVERAGES 1DATATYPE rawSCANCAL -0.10043094726LOWFILTER 500SCANAMP 5.13325516467XSPEED 3.0YCOUNT 100XRANGE -10 – 10 mmYRANGE -10 – 10 mmTime samples 512Time step 100 fsApproximate x count 100

Table 6.3: Teraview acquisition parameters.

phantom side of the TPX slide, consisting of some combination of TPX/air, TPX/sticker,

TPX/paint, sticker/air, sticker/paint, and paint/air interfaces, depending on the phantom

and pixel. Obviously there are no phantom or pixel specific data in the time-series until

the terahertz radiation has had time to travel through the TPX, reflect off the interesting

interfaces, and return back through the TPX. In practice, this ‘non-interesting’ section

represented more than half of the time series, and so for the purposes of this study the first

50% of each pulse was windowed out before taking time or frequency domain data.

The ground truth for the two sticker based phantoms was calculated by scaling and

registering the optical image of the phantom to a high frequency absorption image using

the edges of the stickers as the surface to be matched. Segmentation was carried out

manually by a single observer.

The manual segmentation of ‘foot’ resulted in 3 regions — TPX (5184 pixels), thick

sticker (820 pixels), and thin sticker (3996 pixels), making a total of 10,000 pixels. The

sticker part was split into thick and thin areas since the outline of the sticker consisted of

110


Figure 6.4: An example terahertz pulse acquired in reflection mode.

thicker plastic than the internal part. The manual segmentation of the ‘rabbit’ results in 6

regions — TPX (3083 pixels), sticker (2066 pixels), green paint (1371 pixels), gold paint

(198 pixels), red paint (3058 pixels), and blue paint (124 pixels), making a total of 9900

pixels. Figure 6.5 shows the results of these manual segmentations.

(a) (b)

Figure 6.5: The results of manual segmentation of the phantoms (a) ‘foot’, and (b) ‘rab-bit’.

The ‘THZ’ phantom was highly speculative and intended as a qualitative test to see

if the hidden paint regions could be distinguished by terahertz imaging. No ground truth

image was therefore produced for this phantom.

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6.2.2 Clustering

In this work I used standard k-means clustering with random initialisation and Euclidean

distance metric. To compensate for the random initialisation the clustering was carried out

between 20 and 30 times for each sample and feature vector, and the results are presented

for the best outcome. The number of clusters to form was chosen based on a priori

information about the constituency of each sample, as given by the number of regions in

manual segmentation — three for the slices of tooth and ‘foot’, and five for ‘rabbit’.

Two approaches were used to form the feature vectors. Firstly, high dimensional vec-

tors were formed based on the entire series from three different domains — the time do-

main, fast Fourier transform (FFT) domain, and the discrete wavelet transform (DWT) [40]

domain. The amplitude of the FFT coefficients describe the spectral power content of the

pulse, and the DWT coefficients provide a combined time and frequency distribution.

Secondly, low dimensional feature vectors were formed by combining parameters de-

rived from the terahertz data which in themselves provide image contrast, such as phase

shifts and the absorbances at various frequencies. For the analyses of the tooth data a

3-dimensional feature vector was built based on the integral phase shift between 0.5 and

1 THz and between 1 THz and 1.5 THz, and the absorbance at 1 THz. For the reflection

data it was necessary to move to lower frequency bands as the frequency range of the

Teraview scanner does not extend as high as that of the Leeds system. The integral phase

shift between 0 THz and 0.5 THz and between 0.5 THz and 1.0 THz, and the absorbance

at 0.8 THz was therefore used instead. The 3-dimensional vectors were then normalised

to be uni-variate within a unit hypercube.

6.2.3 Evaluation

The segmentations were evaluated by counting the number of mis-classified pixels com-

pared with the respective registered ground truths that were previously described. In each

case the results of the best segmentation are presented.

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The reflection data was marred by the occasional missed pixel, an artefact of contin-

uous scanning. If the controlling computer became too busy the data buffer overran and

any pixels scanned in during the busy period were lost. Those pixels were excluded from

the clustering.

6.3 Results

6.3.1 Tooth Slices

A successful segmentation is one that results in three contiguous regions corresponding

to the three regions present in the tooth. It was found that the success of the clustering

was dependent on initialisation, with successful segmentations resulting around 70% of

the time. It was also found that when the segmentations were successful the number of

misclassified pixels varied by less than 1% of the total number of pixels.

Time series FFT DWT Featurecoefficients coefficients vector

Synthetic Tooth 0% 0.04% 0% 1%Slice of Real Tooth 19% 23% 19% 13%

Table 6.4: Number of mis-classified pixels when segmenting a synthetic and a real toothusing k-means clustering on a variety of vectors.

The standard k-means clustering segmented perfectly when using the time-series and

DWT coefficients, with errors present when clustering on Fourier coefficients or the 3D

feature vector (see Table 6.4). Figure 6.6(a-c) shows the best results of the k-means clus-

tering using the various feature vectors. Figure 6.6(d) shows the same parameters and

algorithm as (a) where the segmentation failed due to unfavourable initialisation. In these

images, colour is used to show into which cluster the pixels were allocated, out of the three

possibilities. The colours were manually assigned to the clusters to ease comparison.

The same segmentation techniques were applied to the real slice of tooth, and the

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Figure 6.6: The clustering of the synthetic tooth images using k-means clustering on (a)Time series or DWT coefficients, (b) FFT coefficients, (c) 3D feature vector. (d) shows afailed clustering due to poor initialisation, in this instance on the time series data.

results shown in Figure 6.7. Figure 6.7(a) shows the relative amplitude of the pulse, which

is an indicator of where the boundaries between the different areas lie. Figures 6.7(b)–(d)

show the segmentation results as before, with the hand-segmented boundaries from the

radiograph shown in white.

Figure 6.7: (a) The relative amplitude of the pulse through the real tooth, and the k-meansclustering images on (b) Time series, (c) FFT coefficients, (d) 3D feature vector. Thewhite lines show the boundaries extracted from the hand-segmented radiograph.

6.3.2 Phantoms

The phantoms imaged in reflection mode demonstrated less of a dependence on initiali-

sation than the tooth, in that over 95% of the clusters resulted in groupings that approxi-

mated the ground truth. Successful runs again demonstrated less than 1% variation in the

number of misclassified pixels.

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It was also found that the frequency domain was subject to a lot of noise, below

0.234 THz and above 0.820 THz. To exclude this noise, I used a shortened frequency

vector based on the coefficients only between these frequencies. However, it also found

that the lower frequencies tended to demonstrate noise in the sense that while a given fre-

quency might produce an acceptable image, the frequencies either side were subject to a

lot of noise. The lower frequencies also had a blurring effect due to the longer wavelength.

To compensate for these errors, the lower frequency limit was raised until no further im-

provement in the misclassification rate was obtained. This resulted in a frequency vector

of the amplitude of the eight coefficients between 0.615 THz and 0.820 THz. This was

the same for ‘rabbit’ and ‘foot’.

Phantom Time Frequency DWT Featuredomain domain domain vector

Foot 28.3% 29.6% 28.3% 39.8%Rabbit 50.3% 56.9% 50.2% 64.4%

Table 6.5: Percentage of misclassified pixels for the phantoms imaged in reflection mode,using k-means clustering on three different vectors. In total there were 9,933 ‘foot’ pixelsand 9744 ‘rabbit’ pixels classified.

The results for the ‘foot’ and ‘rabbit’ phantoms are shown in Table 6.5. As can be seen

the percentage errors are far higher for these images than for the tooth, and this is due to

the large influence of the boundaries, and the difficulty in resolving the fine structure of

the stickers at terahertz frequencies. For instance, the smaller toes are less than 1 mm

wide with around 1 mm separation, and yet radiation at 1 THz has a wavelength of 3 mm,

leading inevitably to blurring and diffraction issues. The beam is focused to 0.5 mm at

1 THz which improves the sub-wavelength spatial resolution, however in the case of the

toes the resolution is still not sufficient. Similarly fine structures of the rabbit, like its eye,

nose, and feet, are too small to be resolved by the lower frequencies. Figures 6.8 and 6.9

show examples of the best attempt at segmenting the phantom images using the different

vectors. Note that in these images, the excluded pixels are shown in red. The time domain

115


image of the ‘foot’ particularly demonstrates the boundary effects — the fine detail of the

toe region of the large foot has been entirely lost into one large merged area, as has the

separation between the stickers. By careful selection of higher frequencies, it is possible

to recover some of the toe detail, but the errors remain very large.

(a) (b)

(c) (d)

Figure 6.8: The results of segmenting the ‘foot’ phantom using (a) the time domain, (b)the frequency domain, (c) the DWT domain, and (d) the feature vector.

The challenge of the ‘THZ’ phantom proved too hard for this approach — the differ-

ences between the paints were too small and the lack of uniformity across the phantom

too large for the regions to be distinguishable. An example output of the segmentation

process on this phantom is shown in Figure 6.10, but that picture is typical of the output

from any of the analysis techniques tried.

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(a) (b)

(c) (d)

Figure 6.9: The results of segmenting the ‘rabbit’ phantom using (a) the time domain, (b)the frequency domain, (c) the DWT domain, and (d) the feature vector.

Figure 6.10: The results of segmenting the THZ phantom image using the frequencydomain.

6.4 Discussion

Although speculative, this initial work is promising for transmission modality — k-means

clustering segments the synthetic image without error when using a large feature vector.

117


The segmentation of the real tooth is also promising, although clearly nowhere near as

clean as the synthetic image. This is to be expected, as extra factors such as boundary

effects are now present. A physical examination of the slice of tooth showed that the top

right hand corner of the tooth had been cut more thinly than the rest of the tooth, leading

directly to the difficulties with segmentation in that region, so further quantification of the

segmentation accuracy was not performed.

The poor performance of the phantoms imaged in reflection mode was partially ex-

pected. The phantoms were not designed for this modality — reflection mode imaging is

best suited to thick, layered, samples where the interface between each of the layers pro-

duces a response. The boundary effects were more pronounced for the reflection images,

leading to higher error rates than for the tooth. The buffer overruns for certain pixels led

to entire rows of the image being misaligned, in each case contributing up to 100 pixels

to the error. A further contribution to the error may have been mis-registration with the

ground truth, owing to the difficulty in precisely locating edges. However, the clustering

results were self-consistent, and in the case of the ‘foot’ the segmentation could easily

distinguish between the broad areas of sticker, edge of sticker, and TPX. It was neverthe-

less disappointing that there were 50% or greater errors when classifying ‘rabbit’ pixels.

This again was due to a combination of boundary effects, the inability of terahertz radi-

ation to distinguish between the different paints, and inevitable physical variations in the

application of paint, and the spatial resolution issues.

Although the ‘THZ’ phantom was not expected to be perfectly segmented, least of all

in reflection mode, there was not even any evidence of the different paints used. This is

due to a combination of factors, such as the two paints not being sufficiently distinguish-

able under this frequency of radiation, and varying thickness of paint tending to mask the

effects of different paints. It would have been interesting to have imaged all the phantoms

in transmission mode, but limited machine access did not allow this.

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6.5 Conclusions

The study has demonstrated that a feature based clustering technique has potential value

for the segmentation of transmission images, with the number of groups known a priori

and a random initialisation strategy. Although not the focus of this exploration, existing

initialisation strategies and methods of choosing the number of groups are expected to

remove all the operator intervention, leading to truly unsupervised clustering.

The phantoms were deliberately designed to have a combination of macro- and micro-

structures, in order to assess how well a clustering algorithm could cope with these chal-

lenges. The clustering performed well at the macro structure level, easily distinguishing

between TPX and non-TPX regions in most cases, but the physical limitations of the

phantoms, the technology, and use of reflection imaging meant that the micro structures

could not reliably be extracted.

119

Chapter 7

Conclusions and future work

The work in this thesis was motivated by a desire to deepen our understanding of terahertz

pulsed imaging and provide a range of analysis tools suitable for this domain.

7.1 Summary of Work

In Chapter 4 a model of the noise present in terahertz images was empirically built for

the first time by fitting distributions from the stable family. The resulting models were

statistically shown to accurately model the variation between terahertz pulses acquired

through homogeneous material. Existing denoising techniques were applied in a new

way with extreme denoising in order to quantify the errors introduced and their suitability

for data compression applications.

Chapter 5 presented the novel application of time frequency analysis to terahertz data

in order to estimate the optical properties of the sample. Both the short time Fourier trans-

form and the wide band cross ambiguity functions were shown to be able to distinguish

120

Chapter 7 Conclusions and future work

between nylon and resin, and the short time Fourier transform was additionally used to

calculate the complex refractive index of a number of biomedical samples. A window-

ing technique was also used for the first time to demonstrate the impact that very short

acquisition times would have on complex refractive index calculations.

The focus of Chapter 6 was on segmentation through the original application of an

existing clustering technique to terahertz data by creating feature spaces drawn from the

time domain, frequency domain, joint time frequency domains, and the domain of de-

rived physical characteristics. The techniques were evaluated on both transmission and

reflection mode images.

7.2 Discussion

This thesis set out to provide a number of novel time frequency analyses and analysis

techniques suitable for terahertz pulsed imaging. In Section 1.2 some of the issues facing

terahertz imaging were outlined, and these and other topics of interest are now discussed.

The noise present in terahertz data has been explored in two ways. Firstly it was es-

tablished that the observed inter-pulse variation is not distributed according to a Gaussian

distribution, or a distribution formed from a mixture of Gaussians, except under highly

idealised conditions. This is an important finding, as the standard Gaussian model of noise

is no longer appropriate. Instead, a distribution from the stable family (which includes

Gaussians) must be employed, and the selection of parameters is non-trivial. Although

this is a small step forward in our understanding of terahertz noise, it does pave the way

for better models, better denoising techniques, and algorithms that are more robust against

noise.

The extreme denoising has relevance to the volume of data typically acquired. It was

demonstrated that only 20% of coefficients in the wavelet domain are needed for complex

refractive index calculation. Of course, a truncation scheme such as this does not result

121


in an 80% reduction in storage, since the positions of the untruncated coefficients still

need to be stored. However combining a truncation scheme with a traditional lossless

compression technique would help address this issue.

The issue of long acquisition times was partially dealt with through the examination of

the effect that windowing the pulses has. This windowing is equivalent to the acquisition

of fewer samples per pulse, which in turn directly corresponds to a shorter acquisition

time. As expected, the shorter acquisition times did impact the calculation of complex

refractive index, although the general trend was preserved. This means the acquisition

time to be used depends on the application — if an image is being formed from relative

transmission at one or two frequencies then a very short acquisition may be appropri-

ate. On the other hand, if a detailed analysis of the complex refractive index at terahertz

frequencies is required then longer acquisition times become necessary.

The rest of this work was focused on strategies to manage the high dimensionality of

data in terahertz imaging. The short time Fourier transform looks to be a promising tool

for calculating optical constants, especially for refractive index values of noisy data. In

terahertz imaging one would not normally expect to have homogeneous samples of several

different known thickness, and so in some regards the calculation of optical constants is an

academic exercise. However, characterising these materials is useful for modelling their

responses to terahertz radiation. The context for the extraction of optical constants was

exploring contrast mechanisms, and if a technique such as the STFT can successfully be

used to calculate optical constants, then it should also be able to highlight where materials

of different optical properties are contained within an image. By the same token, the

WBCAF shows potential as a tool to this end, although the inconsistency with traditional

analyses does raise questions about its reliability.

Clustering mechanisms ultimately rely on the optical and physical properties of the

sample being imaged of course, as it is these which cause differences to appear between

the pulses. However, removing the need for a priori information and allowing the com-

122


puter to group together pixels it considers to be similar greatly simplifies the problem.

These techniques show great potential for transmission images, even if questions remain

over the reflection images due to the sub-optimal design of phantom for this modality.

7.3 Future Work

The noise models created using distributions from the stable family form an extremely

good fit to the observed noise. This begs the question of whether there is an underlying

physical process that the stable family is modelling. The suggestion of a relationship be-

tween some of the stable parameters and thickness of material is an interesting avenue for

further research, possibly in terahertz imaging methods such as dark field imaging [44].

These explorations would hopefully result in a better understanding of the noise processes

present, so that better noise models could be built, enabling realistic noise to be added to

synthetic images and more robust algorithms to be developed.

The denoising results suggest great potential for data compression. The basic trunca-

tion scheme used here could potentially be improved through the application of coefficient

quantisation and perhaps some form of linear predictive encoding.

The limitations of an analysis based on the phase of the Fourier transform made as-

sessing the accuracy of the short time Fourier transform difficult. Advances in either

phase unwrapping techniques or the application of methods such as Monte Carlo mod-

elling should provide the ground truth refractive index profile against which the results

presented here could be evaluated. The wide band cross ambiguity analysis results were

not consistent with the results from traditional analyses, and the reasons for this need

further exploration. The obvious question is whether or not this matters if the technique

allows different materials to be distinguished, and this would need further experimenta-

tion.

The time frequency techniques were explored with reflection geometry in mind, the

123


mode in which the real advantage over traditional analyses would be found. The analy-

ses presented would need refining before they are directly applicable to reflection mode

images, however.

Clearly a feature based clustering technique has potential value for transmission im-

ages, as the real slice segmentations show. More work needs to be carried out on the

parameters extracted to form the feature vectors. The absorbance should also be calcu-

lated for a frequency range to provide better robustness to noise. In this study the number

of groups was known a priori, and the issue of initialisation was bypassed by using re-

peated random initialisations. In a complete system, the number of groups to be formed

may not be known, and a better strategy would need to be adopted for initialisation. It

is expected that principal components analysis to be an appropriate tool to try within the

reduced feature spaces, using the Mahalanobis distance metric instead of Euclidean [10].

The development of a segmentation tool in the reflection mode would greatly benefit

from the creation of phantoms better suited to this modality. The phantoms were orig-

inally designed to be imaged in transmission mode, but operational constraints with the

terahertz scanner at Leeds led to them being imaged in reflection mode at Cambridge in-

stead. Similarly, the features on the phantoms were at the limit of the spatial resolution

of terahertz data, causing blurring. It is not known whether the boundary effects are more

pronounced in reflection imaging, and this is a further area for future research.

The errors in the segmentation could be decreased through the application of a spatial

reinforcement strategy, such as relaxation [32], which applies the principle that neigh-

bouring pixels are likely to be of the same material. The application of an expectation

maximisation technique with a hidden ‘thickness’ variable may also decrease the error of

the segmentation of the slice of tooth.

Finally, k-means is just one of a large family of clustering tools, and with more data

there would be increased scope for exploring the use of support vector machines, artificial

neural networks, and other supervised and unsupervised techniques.

124

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132

Appendix A

Probability Plot Correlation Coefficient

Critical Values

A.1 Critical Values

133

Probability Plot Correlation Coefficient Critical Values

N 0.05 0.01 N 0.05 0.01 N 0.05 0.013 0.8687 0.879 36 0.954 0.9686 190 0.9897 0.99274 0.8234 0.8666 37 0.9551 0.9693 200 0.9903 0.9935 0.824 0.8786 38 0.9555 0.97 210 0.9907 0.99336 0.8351 0.888 39 0.9568 0.9704 220 0.991 0.99367 0.8474 0.897 40 0.9576 0.9712 230 0.9914 0.99398 0.859 0.9043 41 0.9589 0.9719 240 0.9917 0.99419 0.8689 0.9115 42 0.9593 0.9723 250 0.9921 0.994310 0.8765 0.9173 43 0.9609 0.973 260 0.9924 0.994511 0.8838 0.9223 44 0.9611 0.9734 270 0.9926 0.994712 0.8918 0.9267 45 0.962 0.9739 280 0.9929 0.994913 0.8974 0.931 46 0.9629 0.9744 290 0.9931 0.995114 0.9029 0.9343 47 0.9637 0.9748 300 0.9933 0.995215 0.908 0.9376 48 0.964 0.9753 310 0.9936 0.995416 0.9121 0.9405 49 0.9643 0.9758 320 0.9937 0.995517 0.916 0.9433 50 0.9654 0.9761 330 0.9939 0.995618 0.9196 0.9452 55 0.9683 0.9781 340 0.9941 0.995719 0.923 0.9479 60 0.9706 0.9797 350 0.9942 0.995820 0.9256 0.9498 65 0.9723 0.9809 360 0.9944 0.995921 0.9285 0.9515 70 0.9742 0.9822 370 0.9945 0.99622 0.9308 0.9535 75 0.9758 0.9831 380 0.9947 0.996123 0.9334 0.9548 80 0.9771 0.9841 390 0.9948 0.996224 0.9356 0.9564 85 0.9784 0.985 400 0.9949 0.996325 0.937 0.9575 90 0.9797 0.9857 450 0.9954 0.996726 0.9393 0.959 95 0.9804 0.9864 500 0.9959 0.99727 0.9413 0.96 100 0.9814 0.9869 550 0.9963 0.997328 0.9428 0.9615 110 0.983 0.9881 600 0.9965 0.997529 0.9441 0.9622 120 0.9841 0.9889 650 0.9968 0.997730 0.9462 0.9634 130 0.9854 0.9897 700 0.997 0.997831 0.9476 0.9644 140 0.9865 0.9904 750 0.9972 0.99832 0.949 0.9652 150 0.9871 0.9909 800 0.9974 0.998133 0.9505 0.9661 160 0.9879 0.9915 850 0.9975 0.998234 0.9521 0.9671 170 0.9887 0.9919 900 0.9977 0.998335 0.953 0.9678 180 0.9891 0.9923 1000 0.9979 0.9984

Table A.1: Critical Values for probability plot correlation coefficients.

134

Appendix B

Full results of STFT Analysis

B.1 Refractive Index Profiles

135


Figure B.1: Refractive index profiles for a resin step wedge using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.02 ps), (b) 10 � 0 � 5 (0.063 ps), (c) 100 (0.2 ps), (d) 100 � 5(0.63 ps), (e) 101 (2 ps), (f) 101 � 5 (6.3 ps), (g) 102 (20 ps), (h) 102 � 5 (63 ps), (i) 103 (200 ps),and (j) 103 � 5 (632 ps).

136


Figure B.2: Refractive index profiles for excised skin using the Gaussian windowed STFTwith widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5 (0.25 ps),(e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps), and (j)103 � 5 (253 ps).

137


Figure B.3: Refractive index profiles for excised fat using the Gaussian windowed STFTwith widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5 (0.25 ps),(e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps), and (j)103 � 5 (253 ps).

138


Figure B.4: Refractive index profiles for excised muscle using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5(0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps),and (j) 103 � 5 (253 ps).

139


Figure B.5: Refractive index profiles for excised vein using the Gaussian windowed STFTwith widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5 (0.25 ps),(e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps), and (j)103 � 5 (253 ps).

140


Figure B.6: Refractive index profiles for excised artery using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5(0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps),and (j) 103 � 5 (253 ps).

141


Figure B.7: Refractive index profiles for excised nerve using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5(0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps),and (j) 103 � 5 (253 ps).

142


B.2 Absorption Coefficient Profiles

Figure B.8: Absorption coefficient profiles for a resin step wedge using the Gaussianwindowed STFT with widths (a) 10 � 1 (0.02 ps), (b) 10 � 0 � 5 (0.063 ps), (c) 100 (0.2 ps), (d)100 � 5 (0.63 ps), (e) 101 (2 ps), (f) 101 � 5 (6.3 ps), (g) 102 (20 ps), (h) 102 � 5 (63 ps), (i) 103

(200 ps), and (j) 103 � 5 (632 ps).

143


Figure B.9: Absorption coefficient profiles for excised skin using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5(0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps),and (j) 103 � 5 (253 ps).

144


Figure B.10: Absorption coefficient profiles for excised fat using the Gaussian windowedSTFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d) 100 � 5(0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103 (80 ps),and (j) 103 � 5 (253 ps).

145


Figure B.11: Absorption coefficient profiles for excised muscle using the Gaussian win-dowed STFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d)100 � 5 (0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103

(80 ps), and (j) 103 � 5 (253 ps).

146


Figure B.12: Absorption coefficient profiles for excised vein using the Gaussian win-dowed STFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d)100 � 5 (0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103

(80 ps), and (j) 103 � 5 (253 ps).

147


Figure B.13: Absorption coefficient profiles for excised artery using the Gaussian win-dowed STFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d)100 � 5 (0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103

(80 ps), and (j) 103 � 5 (253 ps).

148


Figure B.14: Absorption coefficient profiles for excised nerve using the Gaussian win-dowed STFT with widths (a) 10 � 1 (0.008 ps), (b) 10 � 0 � 5 (0.025 ps), (c) 100 (0.08 ps), (d)100 � 5 (0.25 ps), (e) 101 (0.8 ps), (f) 101 � 5 (2.5 ps), (g) 102 (8 ps), (h) 102 � 5 (25 ps), (i) 103

(80 ps), and (j) 103 � 5 (253 ps).

149

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